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 ngon:
It is not easy to draw an ngon, so let's represent the regular polygon or ngon 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 ngon
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
Notice that the pentagon has 5 sides and you can make 5 triangles. Similarly, the ngon 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 ngon
Say that s = 4 and apothem = 8
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 ngon has n triangles
Since there are n triangles in an ngon, 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.
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 ngon
Second generalization of the area of a regular polygon
base = s , height = apothem and the ngon has n sides
Using tan(x) =
s
/
2 × apothem
, we get s = tan(x) × 2 × apothem
Find x for an ngon.
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
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
If we rewrite A = n ×
s × apothem
/
2
it will be easier to manage
A =
n × s^{2}
/
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:
s = 2 × r sin(x)
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) × r^{2}
/
2
Useful trigonometric formula: sin(2x) = 2 sin(x)cos(x)
A = n ×
sin(2x) × r^{2}
/
2

Nov 09, 18 09:40 AM
Learn the three properties of congruence. Examples to illustrate which property.
Read More
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.