Area of a regular polygon

The area of a regular polygon (also called n-gon) can be derived with some basic knowledge of geometry and trigonometry. We will show you how to derive 3 formulas that you can use to get the area of an n-gon.

It is not easy to draw an n-gon, so let's represent the regular polygon or n-gon with a pentagon


Our strategy before we derive the formula of the area of a regular polygon will be to show you how to get the area of a pentagon and then generalize the approach for an n-gon

To get the area of the pentagon above, follow the steps below:

1. Break the pentagon into 5 congruent triangles

2. Get the area for one triangle

3. Multiply by 5 to get the area for all 5 triangles or the whole pentagon



Looking at the triangle above, you can see that the base is s and s is also the length of one side of the pentagon. The red line is the height and it is called apothem in an n-gon

Say that s = 4 and apothem = 8




First generalization of the area of a regular polygon

base = s and height = apothem. An n-gon has n triangles



Sometimes, s is not given, but you know the apothem and the number of sides. So you need to find s in this case. You will need some basic trigonometric identities

You need to know also how to get the central angle. Take a look at the figure again, the central angle is the one in black

Find the area when n = 5 and apothem = 8

Call the angle in orange x and use trigonometric identity.







Now, we have to find x. x is half the angle in black. The angle in black is the central angle.



n is the number of sides.



The angle in orange or x is equal to 72 divided by 2 or 36 degrees. Thus, s = tan(36°) × 16



This math can be done fairly easily, but we will not do it so you can better see how we can generalize and come up with a formula for an n-gon

Second generalization of the area of a regular polygon

base = s , height = apothem and the n-gon has n sides




Find x for an n-gon.











Common pitfall: Thinking that the apothem needs to be written only once! Take a look again. The apothem appears in s and also appears in A

Derivation of the area of a regular polygon when s and n are given, but the apothem is not known

Since the apothem is missing we can use the formula s = tan(x) × 2 × apothem and solve for apothem.

Replace the apothem into the formula for the area and simplify. You will end up with an equation in terms of s and the angle

s = 2 × tan(x) × apothem

A = n × s / 2 ×

s / 2 × tan(x)

Derivation of the area of a regular polygon when the radius is given, but the apothem and s are not known

The radius is the blue line or the hypotenuse. If x is the angle in orange and let the radius be r we get:





s = 2 × r sin(x)




apothem = cos(x) × r







Useful trigonometric formula: sin(2x) = 2 sin(x)cos(x)

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