Home > Geometry > Angles with Triangles and Polygons

Is there a formula for the area of a regular polygon of side #a# having #n# sides?

Answer 1

#S=(n a^2)/4cot(pi/n)#

A regular polygon has all sides equal #a#. Each side is observed from the center #O# at the same angle #phi#. The polygon area is covered by as many isosceles triangles with vertices centered, as sides it has. A regular polygon has a circumscribed circle with radius #r#. Considering now the radius #r# as the side of isosceles triangle with vertice at the center and #a# as third side we have the area #s#

#s = 1/2a r cos(phi/2)#

if we have #n# sides then #phi=(2pi)/n#

and the polygon area is given by

#S = n s = (nar)/2cos(pi/n)#

but #a = 2rsin(phi/2)=2rsin(pi/n)#

now substituting #r = a/(2sin(pi/n))# into the #S# relationship

#S=(n a^2)/4cot(pi/n)#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Parker Allen

Yes, the formula for the area ( A ) of a regular polygon with side length ( a ) and ( n ) sides is:

[ A = \frac{n \times a^2}{4 \times \tan\left(\frac{\pi}{n}\right)} ]

Where:

( n ) is the number of sides,
( a ) is the length of each side, and
( \tan ) denotes the tangent function.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.