$ABCDEFGH$ is a regular octagon of side 12cm. Find the area in square centimeters of trapezoid $BCDE$. Express your answer in simplest radical form.
Call the intersection of BE with CH point X. Then the triangle BCX is a right isosceles right triangle.
This means that angle CBE is a 45 degree angle.
And, both BX and CX are equal to 6·sqrt(2).
CX is a height of the trapezoid and equals 6·sqrt(2).
CD is one base of the trapezoid and equals 12.
BE is the other base of the trapezoid and equals 6·sqrt(2) + 12 + 6·sqrt(2) = 12 + 12·sqrt(2).
The area of the trapezoid is (1/2) · height · (base #1 + base #2)
= (1/2) · 6·sqrt(2) · ( 12 + 12·sqrt(2) )
= 3 ·sqrt(2) · ( 12 + 12·sqrt(2) )
= 36·sqrt(2) + 72