Regular Polygon Area Formula - Math Open Reference

Math Open Reference Home Contact About Subject Index 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.

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 tan is the tangent function calculated in degrees (see Trigonometry Overview)

To see how this equation is derived, see Derivation of regular polygon area formula.

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 sin is the sine function calculated in degrees (see Trigonometry Overview)

To see how this equation is derived, see Derivation of regular polygon area formula.

3. Given the apothem (inradius)

If you know the apothem, or inradius, (the perpendicular distance from center to a side. See figure above), the area is given by: where a is the length of the apothem (inradius) n is the number of sides tan is the tangent function calculated in degrees (see Trigonometry Overview).

To see how this equation is derived, see Derivation of regular polygon area formula.

Irregular Polygons

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

Other polygon topics

General

• Polygon general definition
• Quadrilateral
• Regular polygon
• Irregular polygon
• Convex polygons
• Concave polygons
• Polygon diagonals
• Polygon triangles
• Apothem of a regular polygon
• Polygon center
• Radius of a regular polygon
• Incircle of a regular polygon
• Incenter of a regular polygon
• Circumcircle of a polygon
• Parallelogram inscribed in a quadrilateral

Types of polygon

• Square
• Diagonals of a square
• Rectangle
• Diagonals of a rectangle
• Golden rectangle
• Parallelogram
• Rhombus
• Trapezoid
• Trapezoid median
• Kite
• Inscribed (cyclic) quadrilateral
• Inscribed quadrilateral interior angles
• Inscribed quadrilateral area
• Inscribed quadrilateral diagonals

Area of various polygon types

• Regular polygon area
• Irregular polygon area
• Rhombus area
• Kite area
• Rectangle area
• Area of a square
• Trapezoid area
• Parallelogram area

Perimeter of various polygon types

• Perimeter of a polygon (regular and irregular)
• Perimeter of a triangle
• Perimeter of a rectangle
• Perimeter of a square
• Perimeter of a parallelogram
• Perimeter of a rhombus
• Perimeter of a trapezoid
• Perimeter of a kite

Angles associated with polygons

• Exterior angles of a polygon
• Interior angles of a polygon
• Relationship of interior/exterior angles
• Polygon central angle

Named polygons

• Tetragon, 4 sides
• Pentagon, 5 sides
• Hexagon, 6 sides
• Heptagon, 7 sides
• Octagon, 8 sides
• Nonagon Enneagon, 9 sides
• Decagon, 10 sides
• Undecagon, 11 sides
• Dodecagon, 12 sides

(C) 2011 Copyright Math Open Reference. All rights reserved