Calculate the area of a regular dodecagon with a radius of 5 cm.

1. Problem Description

Calculate the area of a regular dodecagon with a radius of 5 cm.

2. Solution Steps

A regular dodecagon has 12 sides. We can divide it into 12 congruent isosceles triangles, each with its vertex at the center of the dodecagon. The radius of the dodecagon is the length of the two equal sides of each isosceles triangle.

The angle at the center of the dodecagon for each triangle is $360^\circ / 12 = 30^\circ$.

Let $r$ be the radius of the dodecagon (5 cm), and $a$ be the length of the side of the dodecagon. We need to find the area of one triangle, and then multiply by

1

2. The area of a triangle given two sides and the included angle is:

$Area = (1/2) * side1 * side2 * sin(angle)$

In our case, $side1 = side2 = r = 5$ cm, and $angle = 30^\circ$. Therefore, the area of one isosceles triangle is:

$Area_{triangle} = (1/2) * 5 * 5 * sin(30^\circ) = (1/2) * 25 * (1/2) = 25/4 = 6.25$ cm$^2$.

Since there are 12 triangles, the total area of the dodecagon is:

$Area_{dodecagon} = 12 * Area_{triangle} = 12 * 6.25 = 75$ cm$^2$.

3. Final Answer

The area of the regular dodecagon is 75 cm$^2$.