000950 Comprehensive Guide to Polygon Classifications and Types
000950 Comprehensive Guide to Polygon Classifications and Types
0001.
0001.
Polygons
Polygons
0001.0001.
0001.0001.
Classification A
Classification A
0001.0001.0001.
0001.0001.0001.
Complex Polygon
Complex Polygon
0001.0001.0001.0001.
0001.0001.0001.0001.
A complex polygon is a polygon that intersects itself. These types of polygons are also called "self-intersecting polygons." They can take various forms such as a star shape. Calculating the area of a complex polygon can be difficult because the polygon overlaps itself. In some cases, methods like triangulation can be used to break the polygon into simpler non-overlapping regions.
A complex polygon is a polygon that intersects itself. These types of polygons are also called "self-intersecting polygons." They can take various forms such as a star shape. Calculating the area of a complex polygon can be difficult because the polygon overlaps itself. In some cases, methods like triangulation can be used to break the polygon into simpler non-overlapping regions.
0001.0001.0002.
0001.0001.0002.
Simple Polygon
Simple Polygon
0001.0001.0002.0001.
0001.0001.0002.0001.
A simple polygon does not intersect itself. It has a well-defined interior and exterior, and its boundary does not cross itself. Simple polygons can have a variety of side counts and can be either concave or convex. The area of a simple polygon can be calculated using the formula for regular or irregular polygons depending on its side lengths and angle measurements.
A simple polygon does not intersect itself. It has a well-defined interior and exterior, and its boundary does not cross itself. Simple polygons can have a variety of side counts and can be either concave or convex. The area of a simple polygon can be calculated using the formula for regular or irregular polygons depending on its side lengths and angle measurements.
0001.0002.
0001.0002.
Classification B
Classification B
0001.0002.0001.
0001.0002.0001.
Concave Polygon
Concave Polygon
0001.0002.0001.0001.
0001.0002.0001.0001.
A concave polygon has at least one interior angle greater than 180 degrees. Concave polygons can be regular or irregular in shape. Because part of the polygon is "pushed inward," calculating the area of a concave polygon may involve splitting it into smaller convex polygons or using advanced methods. The standard area calculation formulas may not apply directly.
A concave polygon has at least one interior angle greater than 180 degrees. Concave polygons can be regular or irregular in shape. Because part of the polygon is "pushed inward," calculating the area of a concave polygon may involve splitting it into smaller convex polygons or using advanced methods. The standard area calculation formulas may not apply directly.
0001.0002.0002.
0001.0002.0002.
Convex Polygon
Convex Polygon
0001.0002.0002.0001.
0001.0002.0002.0001.
A convex polygon has all interior angles less than 180 degrees. Convex polygons are simpler to handle mathematically because their vertices always point outward, and no part of the shape is "caved in." The area and perimeter of convex polygons can be calculated using known side lengths and trigonometric relationships.
A convex polygon has all interior angles less than 180 degrees. Convex polygons are simpler to handle mathematically because their vertices always point outward, and no part of the shape is "caved in." The area and perimeter of convex polygons can be calculated using known side lengths and trigonometric relationships.
0001.0003.
0001.0003.
Classification C
Classification C
0001.0003.0001.
0001.0003.0001.
Irregular Polygon
Irregular Polygon
0001.0003.0001.0001.
0001.0003.0001.0001.
An irregular polygon has sides of unequal length and angles that are not equal. In real-world applications, irregular polygons are common in nature and in areas like engineering and architecture. The area of an irregular polygon can be calculated using methods such as the Shoelace Theorem, which involves the coordinates of its vertices, or by dividing it into triangles.
An irregular polygon has sides of unequal length and angles that are not equal. In real-world applications, irregular polygons are common in nature and in areas like engineering and architecture. The area of an irregular polygon can be calculated using methods such as the Shoelace Theorem, which involves the coordinates of its vertices, or by dividing it into triangles.
0001.0003.0002.
0001.0003.0002.
Regular Polygon
Regular Polygon
0001.0003.0002.0001.
0001.0003.0002.0001.
A regular polygon has equal side lengths and equal interior angles. It is symmetrical, and its area and perimeter can be calculated using specific formulas. Regular polygons are frequently used in design, architecture, and engineering because of their symmetry and uniformity.
A regular polygon has equal side lengths and equal interior angles. It is symmetrical, and its area and perimeter can be calculated using specific formulas. Regular polygons are frequently used in design, architecture, and engineering because of their symmetry and uniformity.
Perimeter = n * s, where n is the number of sides, and s is the length of each side.
Perimeter = n * s, where n is the number of sides, and s is the length of each side.
The area of a regular polygon can be calculated as:
The area of a regular polygon can be calculated as:
Area = (n * s * a) / 2, where “a” is the apothem, the perpendicular distance from the center to a side.
Area = (n * s * a) / 2, where “a” is the apothem, the perpendicular distance from the center to a side.
0001.0004.
0001.0004.
Classification D (Combined Classification)
Classification D (Combined Classification)
0001.0004.0001.
0001.0004.0001.
Complex Irregular Polygon
Complex Irregular Polygon
0001.0004.0001.0001.
0001.0004.0001.0001.
A complex irregular polygon is both self-intersecting and has sides of unequal length. Calculating the area of such polygons can be very challenging, and it usually requires specialized algorithms or geometric methods.
A complex irregular polygon is both self-intersecting and has sides of unequal length. Calculating the area of such polygons can be very challenging, and it usually requires specialized algorithms or geometric methods.
0001.0004.0002.
0001.0004.0002.
Complex Regular Polygon
Complex Regular Polygon
0001.0004.0002.0001.
0001.0004.0002.0001.
A complex regular polygon is self-intersecting but has equal side lengths. The area calculation may involve splitting the polygon into simpler regions or using trigonometric approaches.
A complex regular polygon is self-intersecting but has equal side lengths. The area calculation may involve splitting the polygon into simpler regions or using trigonometric approaches.
0001.0005.
0001.0005.
Classification E (By Number of Sides) n sides = n-gon
Classification E (By Number of Sides) n sides = n-gon
0001.0005.0001.
0001.0005.0001.
Three Sides
Three Sides
0001.0005.0001.0001.
0001.0005.0001.0001.
Triangles
Triangles
0001.0005.0001.0001.0001.
0001.0005.0001.0001.0001.
Types of Triangles
Types of Triangles
0001.0005.0001.0001.0001.0001.
0001.0005.0001.0001.0001.0001.
Scalene Triangle
Scalene Triangle
0001.0005.0001.0001.0001.0001.0001.
0001.0005.0001.0001.0001.0001.0001.
A scalene triangle has sides of different lengths, and its angles are also unequal. The area of a scalene triangle can be calculated using Heron's formula:
A scalene triangle has sides of different lengths, and its angles are also unequal. The area of a scalene triangle can be calculated using Heron's formula:
s = (a + b + c) / 2, where a, b, and c are the side lengths.
s = (a + b + c) / 2, where a, b, and c are the side lengths.
Area = sqrt(s * (s - a) * (s - b) * (s - c)).
Area = sqrt(s * (s - a) * (s - b) * (s - c)).
0001.0005.0001.0001.0001.0002.
0001.0005.0001.0001.0001.0002.
Equilateral Triangle
Equilateral Triangle
0001.0005.0001.0001.0001.0002.0001.
0001.0005.0001.0001.0001.0002.0001.
An equilateral triangle has all sides of equal length. The area of an equilateral triangle can be calculated using the formula:
An equilateral triangle has all sides of equal length. The area of an equilateral triangle can be calculated using the formula:
Area = (sqrt(3) / 4) * s * s, where s is the length of one side.
Area = (sqrt(3) / 4) * s * s, where s is the length of one side.
The perimeter of an equilateral triangle is:
The perimeter of an equilateral triangle is:
Perimeter = 3 * s.
Perimeter = 3 * s.
0001.0005.0001.0001.0001.0003.
0001.0005.0001.0001.0001.0003.
Isosceles Triangle
Isosceles Triangle
0001.0005.0001.0001.0001.0003.0001.
0001.0005.0001.0001.0001.0003.0001.
An isosceles triangle has two sides of equal length and two equal angles. The area can be calculated using standard triangle formulas if the height or base is known, or using Heron's formula when all three sides are known.
An isosceles triangle has two sides of equal length and two equal angles. The area can be calculated using standard triangle formulas if the height or base is known, or using Heron's formula when all three sides are known.
0001.0005.0001.0001.0001.0004.
0001.0005.0001.0001.0001.0004.
Right Triangle
Right Triangle
0001.0005.0001.0001.0001.0004.0001.
0001.0005.0001.0001.0001.0004.0001.
A right triangle has one 90-degree angle. The area of a right triangle is:
A right triangle has one 90-degree angle. The area of a right triangle is:
Area = (1 / 2) * base * height.
Area = (1 / 2) * base * height.
The Pythagorean Theorem applies to right triangles:
The Pythagorean Theorem applies to right triangles:
a * a + b * b = c * c, where a and b are the legs of the triangle, and c is the hypotenuse.
a * a + b * b = c * c, where a and b are the legs of the triangle, and c is the hypotenuse.
0001.0005.0001.0001.0002.
0001.0005.0001.0001.0002.
Triangle-Based Geometry Branches
Triangle-Based Geometry Branches
0001.0005.0001.0001.0002.0001.
0001.0005.0001.0001.0002.0001.
Trigonometry
Trigonometry
0001.0005.0001.0001.0002.0001.0001.
0001.0005.0001.0001.0002.0001.0001.
Trigonometry plays a critical role in polygon calculations, helping to determine angles, sides, and areas using various formulas such as the Law of Cosines, Law of Sines, and Haversine formula. Polygons are integral to the design, analysis, and optimization of many systems and applications that shape our modern world.
Trigonometry plays a critical role in polygon calculations, helping to determine angles, sides, and areas using various formulas such as the Law of Cosines, Law of Sines, and Haversine formula. Polygons are integral to the design, analysis, and optimization of many systems and applications that shape our modern world.
0001.0005.0001.0001.0002.0001.0002.
0001.0005.0001.0001.0002.0001.0002.
Trigonometry Types
Trigonometry Types
0001.0005.0001.0001.0002.0001.0002.0001.
0001.0005.0001.0001.0002.0001.0002.0001.
Plane Trigonometry
Plane Trigonometry
0001.0005.0001.0001.0002.0001.0002.0001.0001.
0001.0005.0001.0001.0002.0001.0002.0001.0001.
Plane Trigonometry Concepts
Plane Trigonometry Concepts
0001.0005.0001.0001.0002.0001.0002.0001.0001.0001.
0001.0005.0001.0001.0002.0001.0002.0001.0001.0001.
Trigonometric ratios are used in polygon calculations, especially in triangles. The key trigonometric ratios are:
Trigonometric ratios are used in polygon calculations, especially in triangles. The key trigonometric ratios are:
Sine = Opposite / Hypotenuse
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Tangent = Opposite / Adjacent
Cotangent = 1 / Tangent
Cotangent = 1 / Tangent
Secant = 1 / Cosine
Secant = 1 / Cosine
Cosecant = 1 / Sine
Cosecant = 1 / Sine
These ratios can be used to find unknown side lengths or angles in polygons involving right triangles or regular polygons.
These ratios can be used to find unknown side lengths or angles in polygons involving right triangles or regular polygons.
0001.0005.0001.0001.0002.0001.0002.0002.
0001.0005.0001.0001.0002.0001.0002.0002.
Spherical Trigonometry
Spherical Trigonometry
0001.0005.0001.0001.0002.0001.0001.0002.0001.
0001.0005.0001.0001.0002.0001.0001.0002.0001.
Spherical Trigonometry Concepts
Spherical Trigonometry Concepts
0001.0005.0001.0001.0002.0001.0001.0002.0001.0001.
0001.0005.0001.0001.0002.0001.0001.0002.0001.0001.
Distance Between Two Points with Known Longitudes and Latitudes
Distance Between Two Points with Known Longitudes and Latitudes
The Haversine formula calculates the shortest distance between two points on a sphere using their latitude and longitude coordinates. The formula is as follows:
The Haversine formula calculates the shortest distance between two points on a sphere using their latitude and longitude coordinates. The formula is as follows:
Let lat1, lon1 be the latitude and longitude of the first point, and lat2, lon2 be the latitude and longitude of the second point. The Haversine formula is:
Let lat1, lon1 be the latitude and longitude of the first point, and lat2, lon2 be the latitude and longitude of the second point. The Haversine formula is:
a = sin((lat2 - lat1) / 2) ^ 2 + cos(lat1) * cos(lat2) * sin((lon2 - lon1) / 2) ^ 2,
a = sin((lat2 - lat1) / 2) ^ 2 + cos(lat1) * cos(lat2) * sin((lon2 - lon1) / 2) ^ 2,
c = 2 * atan2(sqrt(a), sqrt(1 - a)),
c = 2 * atan2(sqrt(a), sqrt(1 - a)),
d = R * c, where R is the radius of the Earth (approximately 6371 kilometers).
d = R * c, where R is the radius of the Earth (approximately 6371 kilometers).
This formula is essential for calculating distances on the Earth's surface and can be used in navigation and geography.
This formula is essential for calculating distances on the Earth's surface and can be used in navigation and geography.
0001.0005.0002.
0001.0005.0002.
Four Sides (Quadrilateral)
Four Sides (Quadrilateral)
0001.0005.0002.0001.
0001.0005.0002.0001.
Irregular Quadrilateral
Irregular Quadrilateral
0001.0005.0002.0001.0001.
0001.0005.0002.0001.0001.
An irregular quadrilateral has sides of unequal length. The area of an irregular quadrilateral can be calculated by dividing it into triangles or using the Shoelace Theorem if the coordinates of the vertices are known.
An irregular quadrilateral has sides of unequal length. The area of an irregular quadrilateral can be calculated by dividing it into triangles or using the Shoelace Theorem if the coordinates of the vertices are known.
0001.0005.0002.0002.
0001.0005.0002.0002.
Regular Quadrilateral
Regular Quadrilateral
0001.0005.0002.0002.0001.
0001.0005.0002.0002.0001.
Rectangle
Rectangle
0001.0005.0002.0002.0001.0001.
0001.0005.0002.0002.0001.0001.
A rectangle has two pairs of equal sides and all angles are 90 degrees.
A rectangle has two pairs of equal sides and all angles are 90 degrees.
0001.0005.0002.0002.0001.0001.0001.
0001.0005.0002.0002.0001.0001.0001.
The area of a rectangle is:
The area of a rectangle is:
Area = length * width.
Area = length * width.
0001.0005.0002.0002.0002.
0001.0005.0002.0002.0002.
Square
Square
0001.0005.0002.0002.0002.0001.
0001.0005.0002.0002.0002.0001.
A square has four equal sides and all angles are 90 degrees.
A square has four equal sides and all angles are 90 degrees.
0001.0005.0002.0002.0002.0002.
0001.0005.0002.0002.0002.0002.
The area of a square is:
The area of a square is:
Area = side * side.
Area = side * side.
0001.0005.0002.0002.0003.
0001.0005.0002.0002.0003.
Trapezoid
Trapezoid
0001.0005.0002.0002.0003.0001.
0001.0005.0002.0002.0003.0001.
A trapezoid (also called a trapezium) has two parallel sides of different lengths.
A trapezoid (also called a trapezium) has two parallel sides of different lengths.
0001.0005.0002.0002.0003.0002.
0001.0005.0002.0002.0003.0002.
The area of a trapezoid is:
The area of a trapezoid is:
Area = (1 / 2) * (base1 + base2) * height.
Area = (1 / 2) * (base1 + base2) * height.
0001.0005.0003.
0001.0005.0003.
Five Sides (Pentagon)
Five Sides (Pentagon)
0001.0005.0003.0001.
0001.0005.0003.0001.
The area of a regular pentagon can be calculated using the formula:
The area of a regular pentagon can be calculated using the formula:
Area = (1 / 4) * sqrt(5 * (5 + 2 * sqrt(5))) * side * side.
Area = (1 / 4) * sqrt(5 * (5 + 2 * sqrt(5))) * side * side.
0001.0005.0004.
0001.0005.0004.
Six Sides (Hexagon)
Six Sides (Hexagon)
0001.0005.0004.0001.
0001.0005.0004.0001.
Types of Hexagons
Types of Hexagons
0001.0005.0004.0001.0001.
0001.0005.0004.0001.0001.
Regular Hexagon
Regular Hexagon
0001.0005.0004.0001.0001.0001.
0001.0005.0004.0001.0001.0001.
A regular hexagon has six equal sides and equal interior angles of 120 degrees. Hexagons appear frequently in nature, such as in honeycombs.
A regular hexagon has six equal sides and equal interior angles of 120 degrees. Hexagons appear frequently in nature, such as in honeycombs.
0001.0005.0004.0001.0001.0002.
0001.0005.0004.0001.0001.0002.
Area = (3 * sqrt(3) / 2) * side * side.
Area = (3 * sqrt(3) / 2) * side * side.
Alternatively, if the hexagon is divided into six equilateral triangles, the area can be calculated as:
Alternatively, if the hexagon is divided into six equilateral triangles, the area can be calculated as:
Area = 6 * (sqrt(3) / 4) * side * side.
Area = 6 * (sqrt(3) / 4) * side * side.
0001.0005.0004.0002.
0001.0005.0004.0002.
Irregular Hexagon
Irregular Hexagon
0001.0005.0004.0002.0001.
0001.0005.0004.0002.0001.
An irregular hexagon has sides of unequal lengths and angles that are not all equal. The area can be calculated using methods such as triangulation or the Shoelace Theorem if the coordinates of the vertices are known.
An irregular hexagon has sides of unequal lengths and angles that are not all equal. The area can be calculated using methods such as triangulation or the Shoelace Theorem if the coordinates of the vertices are known.
0001.0005.0005.
0001.0005.0005.
Seven Sides (Heptagon)
Seven Sides (Heptagon)
0001.0005.0005.0001.
0001.0005.0005.0001.
Types of Heptagons
Types of Heptagons
0001.0005.0005.0001.0001.
0001.0005.0005.0001.0001.
Regular Heptagon
Regular Heptagon
0001.0005.0005.0001.0001.0001.
0001.0005.0005.0001.0001.0001.
A regular heptagon has seven equal sides and seven equal angles.
A regular heptagon has seven equal sides and seven equal angles.
0001.0005.0005.0001.0001.0002.
0001.0005.0005.0001.0001.0002.
Area = (7 / 4) * side * side * cot(pi / 7),
Area = (7 / 4) * side * side * cot(pi / 7),
where cot is the cotangent function.
where cot is the cotangent function.
This formula can be simplified using trigonometric values for cotangent when pi / 7 is known.
This formula can be simplified using trigonometric values for cotangent when pi / 7 is known.
0001.0005.0005.0001.0002.
0001.0005.0005.0001.0002.
Irregular Heptagon
Irregular Heptagon
0001.0005.0005.0001.0002.0001.
0001.0005.0005.0001.0002.0001.
An irregular heptagon has seven sides of unequal lengths and unequal interior angles.
An irregular heptagon has seven sides of unequal lengths and unequal interior angles.
0001.0005.0005.0001.0002.0002.
0001.0005.0005.0001.0002.0002.
The area of an irregular heptagon can be calculated using methods such as dividing the shape into triangles or using the Shoelace Theorem.
The area of an irregular heptagon can be calculated using methods such as dividing the shape into triangles or using the Shoelace Theorem.
0001.0005.0006.
0001.0005.0006.
Eight Sides (Octagon)
Eight Sides (Octagon)
0001.0005.0006.0001.
0001.0005.0006.0001.
Types of Octagons
Types of Octagons
0001.0005.0006.0001.0001.
0001.0005.0006.0001.0001.
Regular Octagon
Regular Octagon
0001.0005.0006.0001.0001.0001.
0001.0005.0006.0001.0001.0001.
A regular octagon has eight equal sides and equal interior angles of 135 degrees.
A regular octagon has eight equal sides and equal interior angles of 135 degrees.
0001.0005.0006.0001.0001.0002.
0001.0005.0006.0001.0001.0002.
The area of a regular octagon can be calculated using the following formula:
The area of a regular octagon can be calculated using the following formula:
Area = 2 * (1 + sqrt(2)) * side * side.
Area = 2 * (1 + sqrt(2)) * side * side.
Regular octagons are often used in design and architecture, such as in stop signs or ornamental tiling.
Regular octagons are often used in design and architecture, such as in stop signs or ornamental tiling.
0001.0005.0006.0002.
0001.0005.0006.0002.
Irregular Octagon
Irregular Octagon
0001.0005.0006.0002.0001.
0001.0005.0006.0002.0001.
An irregular octagon has eight sides of different lengths and unequal interior angles. Calculating the area requires dividing the polygon into triangles or using coordinate geometry.
An irregular octagon has eight sides of different lengths and unequal interior angles. Calculating the area requires dividing the polygon into triangles or using coordinate geometry.
0001.0005.0007.
0001.0005.0007.
Nine Sides (Nonagon)
Nine Sides (Nonagon)
0001.0005.0007.0001.
0001.0005.0007.0001.
Types of Nonagon
Types of Nonagon
0001.0005.0007.0001.0001.
0001.0005.0007.0001.0001.
Regular Nonagon
Regular Nonagon
0001.0005.0007.0001.0001.0001.
0001.0005.0007.0001.0001.0001.
A regular nonagon has nine equal sides and equal interior angles.
A regular nonagon has nine equal sides and equal interior angles.
0001.0005.0007.0001.0001.0002.
0001.0005.0007.0001.0001.0002.
The area of a regular nonagon is:
The area of a regular nonagon is:
Area = (9 / 4) * side * side * cot(pi / 9),
Area = (9 / 4) * side * side * cot(pi / 9),
where cot is the cotangent function.
where cot is the cotangent function.
0001.0005.0007.0001.0002.
0001.0005.0007.0001.0002.
Irregular Nonagon
Irregular Nonagon
0001.0005.0007.0001.0002.0001.
0001.0005.0007.0001.0002.0001.
An irregular nonagon has nine sides of unequal lengths and unequal angles.
An irregular nonagon has nine sides of unequal lengths and unequal angles.
0001.0005.0007.0001.0002.0002.
0001.0005.0007.0001.0002.0002.
Its area can be calculated using triangulation, the Shoelace Theorem, or other advanced methods.
Its area can be calculated using triangulation, the Shoelace Theorem, or other advanced methods.
0001.0005.0008.
0001.0005.0008.
Ten Sides (Decagon)
Ten Sides (Decagon)
0001.0005.0008.0001.
0001.0005.0008.0001.
Types of Decagons
Types of Decagons
0001.0005.0008.0001.0001.
0001.0005.0008.0001.0001.
Regular Decagon
Regular Decagon
0001.0005.0008.0001.0001.0001.
0001.0005.0008.0001.0001.0001.
A regular decagon has ten equal sides and equal interior angles.
A regular decagon has ten equal sides and equal interior angles.
0001.0005.0008.0001.0001.0002.
0001.0005.0008.0001.0001.0002.
The formula for the area of a regular decagon is:
The formula for the area of a regular decagon is:
Area = (5 / 2) * side * side * sqrt(5 + 2 * sqrt(5)).
Area = (5 / 2) * side * side * sqrt(5 + 2 * sqrt(5)).
0001.0005.0008.0001.0002.
0001.0005.0008.0001.0002.
Irregular Decagon
Irregular Decagon
0001.0005.0008.0001.0002.0001.
0001.0005.0008.0001.0002.0001.
An irregular decagon has ten sides of unequal lengths and unequal angles
An irregular decagon has ten sides of unequal lengths and unequal angles
0001.0005.0008.0001.0002.0002.
0001.0005.0008.0001.0002.0002.
The area can be calculated by dividing the decagon into triangles or using the Shoelace Theorem with the vertices' coordinates.
The area can be calculated by dividing the decagon into triangles or using the Shoelace Theorem with the vertices' coordinates.
0001.0005.0009.
0001.0005.0009.
Eleven Sides (Nonagon (or Enneagon) )
Eleven Sides (Nonagon (or Enneagon) )
0001.0005.0010.
0001.0005.0010.
Twelve Sides (Dodecagon)
Twelve Sides (Dodecagon)
0001.0005.0010.0001.
0001.0005.0010.0001.
Types of Dodecagons
Types of Dodecagons
0001.0005.0010.0001.0001.
0001.0005.0010.0001.0001.
Regular Dodecagon
Regular Dodecagon
0001.0005.0010.0001.0001.0001.
0001.0005.0010.0001.0001.0001.
A regular dodecagon has twelve equal sides and equal interior angles.
A regular dodecagon has twelve equal sides and equal interior angles.
0001.0005.0010.0001.0001.0002.
0001.0005.0010.0001.0001.0002.
The formula for the area of a regular dodecagon is:
The formula for the area of a regular dodecagon is:
Area = 3 * side * side * (2 + sqrt(3)).
Area = 3 * side * side * (2 + sqrt(3)).
0001.0005.0010.0001.0002.
0001.0005.0010.0001.0002.
Irregular Dodecagon
Irregular Dodecagon
0001.0005.0010.0001.0002.0001.
0001.0005.0010.0001.0002.0001.
An irregular dodecagon has twelve sides of different lengths and unequal angles.
An irregular dodecagon has twelve sides of different lengths and unequal angles.
0001.0005.0010.0001.0002.0002.
0001.0005.0010.0001.0002.0002.
The area can be computed using triangulation, the Shoelace Theorem, or other geometric methods.
The area can be computed using triangulation, the Shoelace Theorem, or other geometric methods.
0001.0005.0011.
0001.0005.0011.
Thirteen Sides (Triskaidecagon (13-gon))
Thirteen Sides (Triskaidecagon (13-gon))
0001.0005.0011.0001.
0001.0005.0011.0001.
The triskaidecagon is a polygon with thirteen sides.
The triskaidecagon is a polygon with thirteen sides.
0001.0005.0011.0002.
0001.0005.0011.0002.
The sum of its interior angles is 1980 degrees, and each interior angle in a regular triskaidecagon is approximately 152.31 degrees.
The sum of its interior angles is 1980 degrees, and each interior angle in a regular triskaidecagon is approximately 152.31 degrees.
0001.0005.0012.
0001.0005.0012.
Fourteen Sides Tetrakaidecagon (14-gon))
Fourteen Sides Tetrakaidecagon (14-gon))
0001.0005.0012.0001.
0001.0005.0012.0001.
A tetrakaidecagon is a fourteen-sided polygon. Its interior angle sum is 2160 degrees, and each angle in a regular tetrakaidecagon measures about 154.29 degrees.
A tetrakaidecagon is a fourteen-sided polygon. Its interior angle sum is 2160 degrees, and each angle in a regular tetrakaidecagon measures about 154.29 degrees.
0001.0005.0013.
0001.0005.0013.
Fifteen Sides (Pentadecagon (15-gon))
Fifteen Sides (Pentadecagon (15-gon))
0001.0005.0013.0001.
0001.0005.0013.0001.
A polygon with fifteen sides, the pentadecagon has an interior angle sum of 2340 degrees. In a regular pentadecagon, each angle is 156 degrees.
A polygon with fifteen sides, the pentadecagon has an interior angle sum of 2340 degrees. In a regular pentadecagon, each angle is 156 degrees.
0001.0005.0014.
0001.0005.0014.
Sixteen Sides (Hexakaidecagon (16-gon))
Sixteen Sides (Hexakaidecagon (16-gon))
0001.0005.0014.0001.
0001.0005.0014.0001.
The hexakaidecagon has sixteen sides. Its interior angle sum is 2520 degrees, with each interior angle measuring about 157.5 degrees in a regular hexakaidecagon.
The hexakaidecagon has sixteen sides. Its interior angle sum is 2520 degrees, with each interior angle measuring about 157.5 degrees in a regular hexakaidecagon.
0001.0005.0015.
0001.0005.0015.
Seventeen Sides (Heptadecagon (17-gon))
Seventeen Sides (Heptadecagon (17-gon))
0001.0005.0015.0001.
0001.0005.0015.0001.
A heptadecagon is a seventeen-sided polygon with an interior angle sum of 2700 degrees. Each angle in a regular heptadecagon is approximately 158.82 degrees.
A heptadecagon is a seventeen-sided polygon with an interior angle sum of 2700 degrees. Each angle in a regular heptadecagon is approximately 158.82 degrees.
0001.0005.0016.
0001.0005.0016.
Eighteen Sides (Octakaidecagon (18-gon))
Eighteen Sides (Octakaidecagon (18-gon))
0001.0005.0016.0001.
0001.0005.0016.0001.
The octakaidecagon has eighteen sides and an interior angle sum of 2880 degrees. In a regular octakaidecagon, each angle measures about 160 degrees.
The octakaidecagon has eighteen sides and an interior angle sum of 2880 degrees. In a regular octakaidecagon, each angle measures about 160 degrees.
0001.0005.0017.
0001.0005.0017.
Nineteen Sides (Enneadecagon (19-gon))
Nineteen Sides (Enneadecagon (19-gon))
0001.0005.0017.0001.
0001.0005.0017.0001.
An enneadecagon is a nineteen-sided polygon. The sum of its interior angles is 3060 degrees, and each angle in a regular enneadecagon is approximately 161.05 degrees.
An enneadecagon is a nineteen-sided polygon. The sum of its interior angles is 3060 degrees, and each angle in a regular enneadecagon is approximately 161.05 degrees.
0001.0005.0018.
0001.0005.0018.
Twenty Sides (Icosagon (20-gon))
Twenty Sides (Icosagon (20-gon))
0001.0005.0018.0001.
0001.0005.0018.0001.
An icosagon has twenty sides. Its interior angle sum is 3240 degrees, with each interior angle measuring 162 degrees in a regular icosagon.
An icosagon has twenty sides. Its interior angle sum is 3240 degrees, with each interior angle measuring 162 degrees in a regular icosagon.
0001.0005.0019.
0001.0005.0019.
Thirty Sides (Triacontagon (30-gon))
Thirty Sides (Triacontagon (30-gon))
0001.0005.0019.0001.
0001.0005.0019.0001.
The triacontagon is a thirty-sided polygon. Its interior angle sum is 5040 degrees. Each angle in a regular triacontagon is 168 degrees.
The triacontagon is a thirty-sided polygon. Its interior angle sum is 5040 degrees. Each angle in a regular triacontagon is 168 degrees.
0001.0005.0020.
0001.0005.0020.
Forty Sides (Tetracontagon (40-gon))
Forty Sides (Tetracontagon (40-gon))
0001.0005.0020.0001.
0001.0005.0020.0001.
A tetracontagon has forty sides. The sum of its interior angles is 7200 degrees, and in a regular tetracontagon, each angle is 171 degrees.
A tetracontagon has forty sides. The sum of its interior angles is 7200 degrees, and in a regular tetracontagon, each angle is 171 degrees.
0001.0005.0021.
0001.0005.0021.
Fifty Sides (Pentacontagon (50-gon))
Fifty Sides (Pentacontagon (50-gon))
0001.0005.0021.0001.
0001.0005.0021.0001.
A polygon with fifty sides, the pentacontagon has an interior angle sum of 9000 degrees. Each interior angle measures 172.8 degrees in a regular pentacontagon.
A polygon with fifty sides, the pentacontagon has an interior angle sum of 9000 degrees. Each interior angle measures 172.8 degrees in a regular pentacontagon.
0001.0005.0022.
0001.0005.0022.
Sixty Sides (Hexacontagon (60-gon))
Sixty Sides (Hexacontagon (60-gon))
0001.0005.0022.0001.
0001.0005.0022.0001.
The hexacontagon has sixty sides and an interior angle sum of 10800 degrees. In a regular hexacontagon, each angle measures 174 degrees.
The hexacontagon has sixty sides and an interior angle sum of 10800 degrees. In a regular hexacontagon, each angle measures 174 degrees.
0001.0005.0023.
0001.0005.0023.
Seventy Sides (Heptacontagon (70-gon))
Seventy Sides (Heptacontagon (70-gon))
0001.0005.0023.0001.
0001.0005.0023.0001.
A heptacontagon is a polygon with seventy sides. Its interior angle sum is 12600 degrees, with each angle measuring approximately 174.86 degrees in a regular heptacontagon.
A heptacontagon is a polygon with seventy sides. Its interior angle sum is 12600 degrees, with each angle measuring approximately 174.86 degrees in a regular heptacontagon.
0001.0005.0024.
0001.0005.0024.
Eighty Sides (Octacontagon (80-gon))
Eighty Sides (Octacontagon (80-gon))
0001.0005.0024.0001.
0001.0005.0024.0001.
An octacontagon has eighty sides. The sum of its interior angles is 14400 degrees, and each interior angle in a regular octacontagon measures 175.5 degrees.
An octacontagon has eighty sides. The sum of its interior angles is 14400 degrees, and each interior angle in a regular octacontagon measures 175.5 degrees.
0001.0005.0025.
0001.0005.0025.
Ninety Sides (Enneacontagon (90-gon))
Ninety Sides (Enneacontagon (90-gon))
0001.0005.0025.0001.
0001.0005.0025.0001.
The enneacontagon has ninety sides. Its interior angle sum is 16200 degrees, with each angle in a regular enneacontagon measuring approximately 176 degrees.
The enneacontagon has ninety sides. Its interior angle sum is 16200 degrees, with each angle in a regular enneacontagon measuring approximately 176 degrees.
0001.0005.0026.
0001.0005.0026.
One Hundred Sides (Hectogon (100-gon))
One Hundred Sides (Hectogon (100-gon))
0001.0005.0026.0001.
0001.0005.0026.0001.
A hectogon is a polygon with one hundred sides. Its interior angle sum is 18000 degrees, and each angle in a regular hectogon measures 176.4 degrees.
A hectogon is a polygon with one hundred sides. Its interior angle sum is 18000 degrees, and each angle in a regular hectogon measures 176.4 degrees.
0001.0005.0027.
0001.0005.0027.
One Thousand Sides (Chiliagon (1,000-gon))
One Thousand Sides (Chiliagon (1,000-gon))
0001.0005.0027.0001.
0001.0005.0027.0001.
A chiliagon is a polygon with one thousand sides. Its interior angle sum is 179640 degrees, and each angle in a regular chiliagon is approximately 179.64 degrees.
A chiliagon is a polygon with one thousand sides. Its interior angle sum is 179640 degrees, and each angle in a regular chiliagon is approximately 179.64 degrees.
0001.0005.0028.
0001.0005.0028.
Ten Thousand Sides (Myriagon (10,000-gon))
Ten Thousand Sides (Myriagon (10,000-gon))
0001.0005.0028.0001.
0001.0005.0028.0001.
A myriagon has ten thousand sides. The sum of its interior angles is 17996400 degrees, with each angle in a regular myriagon being approximately 179.964 degrees.
A myriagon has ten thousand sides. The sum of its interior angles is 17996400 degrees, with each angle in a regular myriagon being approximately 179.964 degrees.
0001.0005.0029.
0001.0005.0029.
One Million Sides (Megagon (1,000,000-gon))
One Million Sides (Megagon (1,000,000-gon))
0001.0005.0029.0001.
0001.0005.0029.0001.
A megagon is a polygon with one million sides. The interior angle sum is approximately 179999640 degrees, with each angle in a regular megagon approaching 179.999964 degrees.
A megagon is a polygon with one million sides. The interior angle sum is approximately 179999640 degrees, with each angle in a regular megagon approaching 179.999964 degrees.
0001.0005.0030.
0001.0005.0030.
10^100 Sides (Googolgon (10^100-gon))
10^100 Sides (Googolgon (10^100-gon))
0001.0005.0030.0001.
0001.0005.0030.0001.
A googolgon is a polygon with googol (10^100) sides. Its interior angles are almost indistinguishable from a circle, making each angle approach 180 degrees.
A googolgon is a polygon with googol (10^100) sides. Its interior angles are almost indistinguishable from a circle, making each angle approach 180 degrees.
0001.0005.0031.0001.
0001.0005.0031.0001.
m Sides = m-gon (m = infinity) (Circle)
m Sides = m-gon (m = infinity) (Circle)
0001.0005.0031.0001.0001.
0001.0005.0031.0001.0001.
A circle is the limit of an m-gon as the number of sides approaches infinity. Its interior angles approach 180 degrees.
A circle is the limit of an m-gon as the number of sides approaches infinity. Its interior angles approach 180 degrees.
0001.0005.0031.0002.
0001.0005.0031.0002.
Ellipse
Ellipse
0001.0005.0031.0002.0001.
0001.0005.0031.0002.0001.
An ellipse is a generalization of a circle, defined by two foci. While not a polygon, it is closely related to the concept of infinite-sided polygons.
An ellipse is a generalization of a circle, defined by two foci. While not a polygon, it is closely related to the concept of infinite-sided polygons.
0001.0005.0031.0003.
0001.0005.0031.0003.
Sector
Sector
0001.0005.0031.0003.0001.
0001.0005.0031.0003.0001.
A sector is a portion of a circle defined by two radii and the arc between them.
A sector is a portion of a circle defined by two radii and the arc between them.
0002.
0002.
Applications of Polygons in Various Fields
Applications of Polygons in Various Fields
0002.0001.
0002.0001.
Polygons play a significant role in various fields of study and practical applications. Some of the key areas include architecture, engineering, computer graphics, and geographical mapping. Understanding the properties and classifications of polygons helps in designing structures, visualizing objects, and creating efficient algorithms.
Polygons play a significant role in various fields of study and practical applications. Some of the key areas include architecture, engineering, computer graphics, and geographical mapping. Understanding the properties and classifications of polygons helps in designing structures, visualizing objects, and creating efficient algorithms.
0002.0001.0001.
0002.0001.0001.
Architecture and Engineering
Architecture and Engineering
0002.0001.0001.0001.
0002.0001.0001.0001.
In architecture and civil engineering, polygons are fundamental in designing structures, facades, and floor plans. Regular polygons, such as hexagons and octagons, are often used in tiling, domes, and roofing patterns due to their symmetry and aesthetic appeal. The strength and efficiency of polygons, especially triangles and hexagons, are utilized in the construction of bridges, towers, and geodesic domes. Irregular polygons are also used in the design of roads, parking lots, and other non-symmetrical layouts.
In architecture and civil engineering, polygons are fundamental in designing structures, facades, and floor plans. Regular polygons, such as hexagons and octagons, are often used in tiling, domes, and roofing patterns due to their symmetry and aesthetic appeal. The strength and efficiency of polygons, especially triangles and hexagons, are utilized in the construction of bridges, towers, and geodesic domes. Irregular polygons are also used in the design of roads, parking lots, and other non-symmetrical layouts.
0002.0001.0002.
0002.0001.0002.
Structural Engineering
Structural Engineering
0002.0001.0002.0001.
0002.0001.0002.0001.
In structural engineering, geometry and triangles play an essential role in designing and analyzing various structures. The triangle is considered the strongest geometric shape because it cannot easily change shape without altering the length of its sides. This property makes triangles a fundamental component in structures such as bridges, roofs, and towers, where stability and strength are paramount.
In structural engineering, geometry and triangles play an essential role in designing and analyzing various structures. The triangle is considered the strongest geometric shape because it cannot easily change shape without altering the length of its sides. This property makes triangles a fundamental component in structures such as bridges, roofs, and towers, where stability and strength are paramount.
Trusses, which consist of triangular units connected at joints, are widely used in structural engineering to distribute forces and provide support over long spans. By using trusses, engineers can reduce or eliminate bending moment forces and shear forces. Bending moments occur when an external force causes a structural element to bend, while shear forces act perpendicular to the element’s length. These forces can lead to structural failures if not managed properly.
Trusses, which consist of triangular units connected at joints, are widely used in structural engineering to distribute forces and provide support over long spans. By using trusses, engineers can reduce or eliminate bending moment forces and shear forces. Bending moments occur when an external force causes a structural element to bend, while shear forces act perpendicular to the element’s length. These forces can lead to structural failures if not managed properly.
In a truss, the load is typically distributed evenly among the members, and the forces are primarily carried by axial forces, meaning the members are either in tension (stretched) or compression (squeezed). Tension and compression are considered normal forces, and they act along the axis of the structural element. Because triangles are rigid and stable, trusses help to minimize bending and shear forces, leaving the structure to handle primarily these normal forces.
In a truss, the load is typically distributed evenly among the members, and the forces are primarily carried by axial forces, meaning the members are either in tension (stretched) or compression (squeezed). Tension and compression are considered normal forces, and they act along the axis of the structural element. Because triangles are rigid and stable, trusses help to minimize bending and shear forces, leaving the structure to handle primarily these normal forces.
This design allows for longer spans without the need for additional supports or columns, making trusses ideal for large open spaces such as bridges, auditoriums, and stadiums. The elimination of bending and shear forces in favor of tension and compression allows for more efficient use of materials, reducing the overall weight and cost of the structure while increasing its durability.
This design allows for longer spans without the need for additional supports or columns, making trusses ideal for large open spaces such as bridges, auditoriums, and stadiums. The elimination of bending and shear forces in favor of tension and compression allows for more efficient use of materials, reducing the overall weight and cost of the structure while increasing its durability.
0002.0001.0003.
0002.0001.0003.
Computer Graphics
Computer Graphics
0002.0001.0003.0001.
0002.0001.0003.0001.
In computer graphics and video game development, polygons are used to model three-dimensional objects. The basic building block of 3D models is the triangle (a polygon with three sides), but quadrilaterals and other polygons are also commonly used. The more sides a polygon has, the smoother the object appears. Polygonal meshes (collections of polygons) are used to create the surfaces of objects in 3D environments.
In computer graphics and video game development, polygons are used to model three-dimensional objects. The basic building block of 3D models is the triangle (a polygon with three sides), but quadrilaterals and other polygons are also commonly used. The more sides a polygon has, the smoother the object appears. Polygonal meshes (collections of polygons) are used to create the surfaces of objects in 3D environments.
Rendering techniques such as ray tracing and rasterization rely on polygons to generate realistic images by calculating the interaction of light and geometry. The efficiency of these algorithms is directly tied to the number of polygons used in a scene, making the understanding of polygons crucial in optimizing performance.
Rendering techniques such as ray tracing and rasterization rely on polygons to generate realistic images by calculating the interaction of light and geometry. The efficiency of these algorithms is directly tied to the number of polygons used in a scene, making the understanding of polygons crucial in optimizing performance.
0002.0001.0004.
0002.0001.0004.
Geographical Mapping and Geospatial Analysis
Geographical Mapping and Geospatial Analysis
0002.0001.0004.0001.
0002.0001.0004.0001.
Polygons are used in geographical mapping to represent areas, such as countries, states, cities, and other regions. In geospatial analysis, polygons are used to define boundaries, calculate areas, and analyze spatial relationships between geographic features. Geographic Information Systems (GIS) rely on polygons to represent real-world objects like lakes, forests, and parcels of land.
Polygons are used in geographical mapping to represent areas, such as countries, states, cities, and other regions. In geospatial analysis, polygons are used to define boundaries, calculate areas, and analyze spatial relationships between geographic features. Geographic Information Systems (GIS) rely on polygons to represent real-world objects like lakes, forests, and parcels of land.
The calculation of the area of a polygon based on latitude and longitude coordinates is crucial in applications like land surveying, environmental monitoring, and urban planning. As mentioned earlier, the Haversine formula is used to compute distances between points on the Earth's surface.
The calculation of the area of a polygon based on latitude and longitude coordinates is crucial in applications like land surveying, environmental monitoring, and urban planning. As mentioned earlier, the Haversine formula is used to compute distances between points on the Earth's surface.
0002.0001.0005.
0002.0001.0005.
Robotics and Path Planning
Robotics and Path Planning
0002.0001.0005.0001.
0002.0001.0005.0001.
In robotics, polygons are used in path planning algorithms to navigate through environments with obstacles. The environment is often represented as a series of polygons, and the robot must find the shortest path around obstacles to reach a destination. Algorithms like A* (A-star) and Dijkstra's Algorithm use polygonal representations to compute efficient paths.
In robotics, polygons are used in path planning algorithms to navigate through environments with obstacles. The environment is often represented as a series of polygons, and the robot must find the shortest path around obstacles to reach a destination. Algorithms like A* (A-star) and Dijkstra's Algorithm use polygonal representations to compute efficient paths.
Polygons are also used in computer vision for object recognition, where shapes are analyzed to identify objects in images and videos.
Polygons are also used in computer vision for object recognition, where shapes are analyzed to identify objects in images and videos.
0002.0001.0006.
0002.0001.0006.
Interior Design and Art
Interior Design and Art
0002.0001.0006.0001.
0002.0001.0006.0001.
In interior design and art, polygons are widely used to create patterns and motifs. Geometric shapes such as triangles, squares, hexagons, and octagons are used in tiling patterns, furniture design, and wall decorations. Artists often use polygons to explore concepts of symmetry, balance, and proportion in their works.
In interior design and art, polygons are widely used to create patterns and motifs. Geometric shapes such as triangles, squares, hexagons, and octagons are used in tiling patterns, furniture design, and wall decorations. Artists often use polygons to explore concepts of symmetry, balance, and proportion in their works.
Polygons are also used in mosaic art and stained glass window designs, where regular and irregular polygons are arranged to create intricate visual effects.
Polygons are also used in mosaic art and stained glass window designs, where regular and irregular polygons are arranged to create intricate visual effects.
0002.0001.0007.
0002.0001.0007.
Science and Chemistry
Science and Chemistry
0002.0001.0007.0001.
0002.0001.0007.0001.
In science, polygons appear in the study of molecular structures, where atoms are arranged in specific geometric patterns. For example, benzene molecules are modeled as hexagons due to the arrangement of carbon atoms in a ring. Polygons are also used to represent crystal structures in materials science.
In science, polygons appear in the study of molecular structures, where atoms are arranged in specific geometric patterns. For example, benzene molecules are modeled as hexagons due to the arrangement of carbon atoms in a ring. Polygons are also used to represent crystal structures in materials science.
In chemistry, polygons help visualize molecular bonds and the arrangement of atoms in compounds. Regular polygons are used in organic chemistry to model cyclic compounds, while irregular polygons represent more complex molecular structures.
In chemistry, polygons help visualize molecular bonds and the arrangement of atoms in compounds. Regular polygons are used in organic chemistry to model cyclic compounds, while irregular polygons represent more complex molecular structures.
0002.0001.0008.
0002.0001.0008.
Mathematical Research
Mathematical Research
0002.0001.0008.0001.
0002.0001.0008.0001.
Polygons play a fundamental role in mathematical research, especially in the fields of geometry, trigonometry, and algebra. Mathematicians study the properties of regular and irregular polygons, such as symmetry, angles, and areas, to develop new theorems and solve complex problems.
Polygons play a fundamental role in mathematical research, especially in the fields of geometry, trigonometry, and algebra. Mathematicians study the properties of regular and irregular polygons, such as symmetry, angles, and areas, to develop new theorems and solve complex problems.
Polygons are also used in number theory, where they are related to polygonal numbers. For example, triangular numbers, square numbers, and pentagonal numbers are part of a broader class of figurate numbers, which represent the arrangement of dots or objects in the shape of regular polygons.
Polygons are also used in number theory, where they are related to polygonal numbers. For example, triangular numbers, square numbers, and pentagonal numbers are part of a broader class of figurate numbers, which represent the arrangement of dots or objects in the shape of regular polygons.
0003.
0003.
Conclusion
Conclusion
0003.0001.
0003.0001.
Polygons, both regular and irregular, are essential elements in geometry and various fields of application. From basic triangles and squares to more complex shapes like dodecagons and beyond, polygons provide the foundation for understanding shapes, structures, and patterns. Their use in architecture, engineering, computer graphics, geospatial analysis, and even chemistry demonstrates their versatility and importance. The study of polygons not only enhances our understanding of geometry but also aids in solving practical problems in the real world.
Polygons, both regular and irregular, are essential elements in geometry and various fields of application. From basic triangles and squares to more complex shapes like dodecagons and beyond, polygons provide the foundation for understanding shapes, structures, and patterns. Their use in architecture, engineering, computer graphics, geospatial analysis, and even chemistry demonstrates their versatility and importance. The study of polygons not only enhances our understanding of geometry but also aids in solving practical problems in the real world.
#KDXVAWRkXeTKKDXVAWsZ2hbK