find the are of a regular octagon whose sides are 12cm long.
I strongly suggest you draw the figure and do the trig yourself. This is not very hard to do.
Draw lines from the center of the octagon to each corner. This forms 8 congruent isosceles triangles with apex angles of 45 degrees. Figure the area of one of them and multiply by 8.
If h is the altitude of each of the 8 triangles, note that 6/h = tan 22.5 degrees. That will help you get the area of each.
To find the area of a regular octagon, you can use the formula:
Area = 2 * (1 + √2) * side length^2
In this case, the side length is given as 12 cm. So, let's substitute the value into the formula:
Area = 2 * (1 + √2) * (12 cm)^2
Area = 2 * (1 + √2) * 144 cm^2
Area = (1 + √2) * 288 cm^2
To calculate the exact numerical value, we can use the approximation √2 ≈ 1.414:
Area ≈ (1 + 1.414) * 288 cm^2
Area ≈ 2.414 * 288 cm^2
Area ≈ 695.232 cm^2
Therefore, the approximate area of the regular octagon with side length 12 cm is 695.232 cm^2.