Notice that a regular dodecagon is made of twelve congruent triangles.
≈ 6.6 inches
Practice makes perfect
We are given that a regular dodecagon is 140 square inches and we are asked to find the length of the apothem. Let's call it a.
First, notice that a regular dodecagon is made of twelve congruent triangles.
This means we can evaluate the area of one triangle by dividing the area of a dodecagon, 140, by the number of triangles, 12.
140/12≈ 11.67
The area of each triangle is approximately 11.67 square inches. Since the central angle of a regular dodecagon is 30^(∘), and the apothem of this figure a is the height of a isosceles triangle, we can use the tangent ratio to write an equation. Let s represent the side length.
Now we can write the equation using the tangent ratio and solve it for s.
tan 15^(∘)=12 s/a ⇒ s=2tan 15^(∘) a
Next we will use the fact that the area of a triangle is the half of the product of its height — in our exercise a — and the corresponding side length, s.
A=1/2 a s
Let's substitute 11.67 for A and 2tan15^(∘) a for s.
Next we will take a square root of both sides of the equation. Notice that since s represents the apothem, we will consider only the positive case when taking the square root of a^2.
