find the area of a regular octagon with side length 12 cm
To find the area of a regular octagon, we can divide the octagon into smaller shapes (usually triangles) and then calculate the area of those smaller shapes to find the total area of the octagon.
1. Divide the regular octagon into 8 congruent triangles by drawing lines from each vertex to the center of the octagon.
2. Each of these triangles is an isosceles triangle with two sides of length 12 cm (the side length of the octagon) and a common vertex angle of 45 degrees.
3. To find the area of one of these triangles, we can use the formula for the area of a triangle: Area = 0.5 * base * height.
4. The base of each triangle is 12 cm and the height can be found by using trigonometry. The height of the triangle is given by h = 12 * sin(45 degrees).
5. Plugging in the values, we have Area = 0.5 * 12 * 12 * sin(45 degrees) = 36 * sin(45 degrees) = 36 * √2 / 2 = 18√2 cm^2.
6. Since the octagon is made up of 8 of these triangles, the total area of the regular octagon is 8 * 18√2 = 144√2 cm^2.
Therefore, the area of a regular octagon with side length 12 cm is 144√2 square cm.