Polygon Calculator
Free Polygon Calculator for 2d geometry. Enter values to get step-by-step solutions with formulas and graphs. Enter your values for instant results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Area = (1/2) x Perimeter x Apothem
The area of a regular polygon equals half the product of its perimeter and apothem. The apothem is s / (2 tan(pi/n)) and the circumradius is s / (2 sin(pi/n)), where n is the number of sides and s is the side length.
Worked Examples
Example 1: Regular Hexagon Calculations
Problem:Find the area, perimeter, apothem, and circumradius of a regular hexagon with side length 8 cm.
Solution:Perimeter = 6 x 8 = 48 cm\nApothem = 8 / (2 x tan(pi/6)) = 8 / (2 x 0.5774) = 6.9282 cm\nArea = 0.5 x 48 x 6.9282 = 166.28 cm2\nCircumradius = 8 / (2 x sin(pi/6)) = 8 / 1.0 = 8.0 cm\nInterior angle = (6-2) x 180 / 6 = 120 degrees
Result:Area: 166.28 cm2 | Perimeter: 48 cm | Apothem: 6.93 cm
Example 2: Regular Octagon Properties
Problem:Calculate the properties of a regular octagon with side length 5 cm.
Solution:Perimeter = 8 x 5 = 40 cm\nApothem = 5 / (2 x tan(pi/8)) = 5 / 0.8284 = 6.0355 cm\nArea = 0.5 x 40 x 6.0355 = 120.71 cm2\nCircumradius = 5 / (2 x sin(pi/8)) = 5 / 0.7654 = 6.5328 cm\nDiagonals = 8(8-3)/2 = 20
Result:Area: 120.71 cm2 | Perimeter: 40 cm | Diagonals: 20
Frequently Asked Questions
What is a regular polygon and how is its area calculated?
A regular polygon is a closed two-dimensional shape where all sides are equal in length and all interior angles are equal in measure. Examples include equilateral triangles, squares, regular pentagons, and regular hexagons. The area of a regular polygon is calculated using the formula Area equals one-half times the perimeter times the apothem, where the apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. Alternatively, the area can be computed as (n times s squared) divided by (4 times the tangent of pi over n), where n is the number of sides and s is the side length. This formula works for any regular polygon regardless of the number of sides.
What is the apothem and circumradius of a polygon?
The apothem and circumradius are two important measurements associated with regular polygons. The apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. It is calculated as s divided by two times the tangent of pi over n, where s is the side length and n is the number of sides. The circumradius is the distance from the center of the polygon to any vertex, and it is calculated as s divided by two times the sine of pi over n. The circumradius always equals or exceeds the apothem. As the number of sides increases, the ratio between circumradius and apothem approaches one, which corresponds to the polygon approaching a circle shape.
How do you calculate the interior angles of a polygon?
The sum of interior angles of any polygon with n sides is given by the formula (n minus 2) times 180 degrees. For a regular polygon where all angles are equal, each interior angle equals this sum divided by n. So for a triangle it is 60 degrees, for a square 90 degrees, for a pentagon 108 degrees, for a hexagon 120 degrees, for an octagon 135 degrees, and for a decagon 144 degrees. The exterior angle of a regular polygon is simply 360 divided by n. Notice that interior and exterior angles always sum to 180 degrees. As the number of sides increases toward infinity, the interior angle approaches but never reaches 180 degrees, and the polygon increasingly resembles a circle.
How many diagonals does a polygon have?
The number of diagonals in a polygon with n sides is calculated using the formula n times (n minus 3) divided by 2. A diagonal is a line segment connecting two non-adjacent vertices. A triangle has zero diagonals because every vertex is adjacent to every other vertex. A quadrilateral has 2 diagonals, a pentagon has 5, a hexagon has 9, an octagon has 20, and a decagon has 35 diagonals. The formula derives from the fact that each vertex connects to n minus 3 other non-adjacent vertices through diagonals, giving n times (n minus 3) connections total, but each diagonal is counted twice so we divide by 2. Understanding diagonals is important in computational geometry and triangulation algorithms.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy