Area of a regular polygon


We will show you how to derive 3 formulas that you can use to get the area of a regular polygon also called n-gon:

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


Representation of an n-gon
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 conguent triangles

2. Get the area for one triangle

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

Notice that the pentagon has 5 sides and you can make 5 triangles. Similarly, the n-gon has n sides and you can make n triangles.

The formula to get the area of a triangle is A =
base × height / 2


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

A =
4 × 8 / 2

Since there are 5 triangles in a pentagon A = 5 ×
4 × 8 / 2


First generalization of the area of a regular polygon

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

Since there are n triangles in an n-gon, A = n ×
s × apothem / 2


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.

apothem
apothem
tan (x) =
s / 2 × 8


tan (x) =
s / 16


If 4 =
8 / 2
  then, 8 = 4 × 2
Similarly, if tan(x) =
s / 16
  then, s = tan(x) × 16


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

The central angle can be found by using the formula: Central angle =
360 degrees / n


n is the number of sides.

Central angle =
360 degrees / 5
= 72 degrees


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

Putting it all together, A = 5 ×
tan(36°) × 16 × 8 / 2


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

A = n ×
s × apothem / 2

Using tan(x) =
s / 2 × apothem
 , we get s = tan(x) × 2 × apothem


Find x for an n-gon.

Central angle =
360 degrees / n


Recall though that x is the orange angle, so  
360 degrees / n
  must be divided by 2


This gives x =  
180 degrees / n




s =   tan(
180 degrees / n
)× 2 × apothem


n-gon area formula
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

apothem =
s / 2 tan(x)


A =
s / 2 tan(x)


If we rewrite A = n ×
s × apothem / 2
  it will be easier to manage


A =
n × s / 2
× apothem


A =
n × s / 2
×
s / 2 × tan(x)


A =
n × s2 / 4 × tan(x)


x is still the same angle, so x =  
180 degrees / n


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:



sin(x) =
s / 2 × r


s = 2 × r sin(x)


cos(x) =
apothem / r


apothem = cos(x) × r

Replace the value for the apothem and s into the formula A = n ×
s × apothem / 2


A = n ×
2 × r sin(x) × cos(x) × r / 2


A = n ×
2 × sin(x) × cos(x) × r2 / 2


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

A = n ×
sin(2x) × r2 / 2


x =
180 degrees / 2









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