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Area of a Regular Polygon

Area of a Regular Polygon

The area of a regular polygon is given by the formula below.

area = (½)(apothem)(perimeter)

A regular hexagon with a vertical line segment from its center to the midpoint of the bottom side, labeled "apothem.

 

Several other area formulas are also available.

 

Regular Polygon Formulas

n = number of sides
s = length of a side
r = apothem (radius of inscribed circle)   
R = radius of circumcircle

Regular hexagon with labeled parts: s = side length along top edge, r = apothem (inscribed circle radius), R = circumcircle...

Sum of interior angles = (n – 2)·180°

Interior angle = Formula for interior angle of a regular polygon: ((n − 2) / n) · 180°

Area = (½)nsr

Three equivalent area formulas for a regular polygon: Area = (1/4)ns²cot(180°/n) = nr²tan(180°/n) = (1/2)nR²sin(360°/n)

Formula: r = (1/2) s cot(180°/n), where r is apothem, s is side length, n is number of sides.

Formula: R = (1/2) s csc(180°/n), where R is circumradius, s is side length, n is number of sides.

A regular pentagon (5-sided polygon) with equal sides and angles, shown as a plain white geometric figure.
Regular Pentagon

A regular hexagon with six equal sides and angles, representing a regular polygon shape.
Regular Hexagon

A regular heptagon (7-sided polygon) with equal sides and angles, no labels visible.
Regular Heptagon

A regular octagon (8-sided polygon) with equal sides and equal interior angles, shown as a geometric example of a regular polygon.
Regular Octagon

A regular polygon with 9 sides (nonagon) showing equal sides and angles, no labels.
Regular Nonagon

 

See also

Area of a convex polygon