Answer:
A = (2+√8)L² - 2L, where L is the length of AB
Step-by-step explanation:
Lets call L the length of one side. The octagon can be divided in 8 triangles composed by the vertices of the octagon and the center. The base of each triangle has length L and the height is L/2 + M, with M such that 2M² = L². This is because the center of the octagon is located at middle height of the center sides. You can reach that height by 'travelling' from the bottom side throught one slanted side (which is the hypotenuse of a isosceles rectangle triangle of height M) and then you travel throught a center side half its length. Thus, h = L/2 + L/√2.
The area of each triangle of the octagon therefore is, as a result, h*L/2 = L²/4 + L²/2√2
Hence, the area of the octagon is 8*(L²/4 + L²/2√2) = (2+√8)L². The unshaded region area can be computed by taking this number and substract to it the value of 2L, which is the area of the shaded region (The double of the length of a side)
As a result,
A = (2+√8)L² - 2L
