Given that BDEF is a square and the area of triangle ABC is 1, find the number of square units in the area of the regular octagon.
+128475 Consider triangle ABC...... the interior angle of a regular octagon = (6/8) 180 = 135°
So angles BCA and BAC are supplemental to this = 45°
Thus sides BA = BC
And ABC is a right angle...so BA and BC are legs
And calling the length of these sides, S......we have that
area of ABC = (1/2) (product of legs) = (1/2) S^2
1 = (1/2) S^2
2 = S^2
S = sqrt (2)
Then AC = sqrt [ BA^2 + BC^2 ] = sqrt [ (sqrt (2))^2 + (sqrt (2))^2 ] = sqrt [ 2 + 2] = sqrt 4 = 2
Then the side of the octagon = 2
A formula for the octagon's area = 2 ( 1 + sqrt 2 ) * S^2 =
2 ( 1 + sqrt 2) * 2^2 =
8 ( 1 + sqrt 2) units ^2 ≈ 19.3 units^2
