Regular polygon area formula - Math Open Reference

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Area of a regular polygon

The number of square units it takes to completely fill a regular polygon. Four different ways to calculate the area are given, with a formula for each.

Try this Drag the orange dots on each vertex to resize the polygon. Alter the number of sides. The area will be continuously calculated.

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The formulae below give the area of a regular polygon. Use the one that matches what you are given to start. They assume you know how many sides the polygon has. Most require a certain knowledge of trigonometry (not covered in this volume, but see Trigonometry Overview).

1. Given the length of a side.

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the area is given by the formula:

where
S is the length of any side
N is the number of sides
π is PI, approximately 3.142
TAN is the tangent function calculated in radians (see Trigonometry Overview)

2. Given the radius (circumradius)

If you know the radius (distance from the center to a vertex, see figure above):

where
R is the radius (circumradius)
N is the number of sides
π is PI, approximately 3.142
SIN is the sine function calculated in radians (see Trigonometry Overview)

3. Given the apothem (inradius)

If you know the apothem, or inradius, (the perpendicular distance from center to a side. See figure above)

where
A is the length of the apothem (inradius)
N is the number of sides
TAN is the tangent function calculated in radians (see Trigonometry Overview).

4. Given the apothem and length of a side

If you know the apothem (the perpendicular distance from center to a side. See figure above) and the length of a side, first determine the perimeter by mutiplying the side length by N. The area is given by

where
A is the length of the apothem
P is the perimeter

Irregular Polygons

Finding the area of an irregular polygon is trickier since there are no easy formulas. See Area of an Irregular Polygon

Related polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons