Use the fact that the measure of each internal angle of a regular polygon can be determined using the formula m∠ α = (n-2)180^(∘)n, where n is the number of the sides in the polygon.
See solution.
Practice makes perfect
Let's imagine that we have to find the length of the apothem of a regular polygon with n sides. First, by drawing radii from the center to the vertices of the polygon, we will divide it into n isosceles triangles.
Let's try to find the length of the apothem, which is marked on the diagram as a. The measure of each internal angle of a regular polygon can be determined using the following formula.
m∠ KCM = (n-2)180^(∘)/n
Radii OC and OB, as well as the others, are the bisectors of the corresponding internal angles. Using this fact, we conclude that the measure of ∠ OCK is twice less than m∠ KCM.
m∠ OCK =(n-2)180^(∘)/n/2=(n-2)180^(∘)/2n
Now, let's consider triangle △ OCK, which is a right triangle. It is a special right triangle if one of the following cases is true.
m∠ OCK =(n-2)180^(∘)/2n=30^(∘)
m∠ OCK =(n-2)180^(∘)/2n=45^(∘)
m∠ OCK =(n-2)180^(∘)/2n=60^(∘)
Let's solve each of these equations for n that represents the number of sides the polygon has.
Case
1
2
3
Equation
(n-2)180^(∘)/2n=30^(∘)
(n-2)180^(∘)/2n=45^(∘)
(n-2)180^(∘)/2n=60^(∘)
LHS * 2n=RHS* 2n
(n-2)180^(∘)=60^(∘) n
(n-2)180^(∘)=90^(∘) n
(n-2)180^(∘)=120^(∘) n
Simplify
3(n-2)=n
2(n-2)=n
3(n-2)=2n
Distribute value
3n-6=n
2n-4=n
3n-6=2n
Rearrange
2n=6
n=4
n=6
Simplify Case 1
n=3
n=4
n=6
Type of the polygon
Triangle
Square
Hexagon
For these three cases we can use special triangles to find the apothem of a polygon. However, when n≠ 2, 3, or 6, the considered triangle will not be a special right triangle, and that is why we will not be able to use its properties.
