An artist has been commissioned to make a stained glass window in the shape of a regular octagon. The octagon must fit inside an 20-in. square space. Determine the length of each side of the octagon. Round to the nearest hundredth of an inch

The we know one side of the square is 20 inches, and one side consist of the addition of a length of the octagon, and two sides for a right angle triangle, which has the smaller angles inside as 45, 45 degrees making those two sides equal to each other, So to find the value of x which is labeled as the side of the octagon, we adding the two sides of the right angle triangle with the side of the octagon to equate to 20 inches

Since a side of the square is 20inches

From the diagram, y is labeled as the two equal sides of the right angle triangle

So

y + x + y = 20 (1)

Using Pythagoras theorem for the right angle triangle

y2 + y2 = x2

2y2 = x2

y2 = (x2)/2

y = x/root(2)

So putting this in the previous equation (1)

x/root(2) + x + x/root(2) = 20

2x/root(2) + x = 20

2x/root(2) + xroot(2)/root(2) = 20

x(2 + root(2))/root(2) = 20

x(2 + root(2)) = 20(root(2))

x = (20(root(2)))/(2 + root(2))

= 28.2843/3.4142 = 8.2343

So therefore answer is approximately 8 inches