ABCD is a square with length of each side 1cm. An octagon is formed by lines joining the vertices of the square to the mid points of opposite sides. Find the area of the octagon?

Answer 1

Let the corners of the square be at (0,0),(1,0),(0,1),(1,1)

y=1-2x intersects y=(1-x)/2 at (1/3,1/3)

So, in each of 4 quadrants, the exterior of the octagon is a trapezoid and a triangle

trapezoid has area 3/16
triangle has area 1/48
sum has area 5/24
The 4 have area 5/6
The octagon thus has area 1/6

Or, consider that the octagon has
side = √5/12
apothem = 1/2√5

area = 1/2 * 8√5/12 * 1/2√5 = 1/6