Solving an octagon?

Question

What is the length of a side of a regular octagon if the longest diagonal is 15 centimeters?

Answer 1

Draw the lines from the center of the octagon to 2 adjacent vertices. Then drop the perpendicular down to get 2 right triangles. These each have hypotenuse 7 1/2, and angle 45/2.

Use the half angle formula to get sin(45/2). Then, the opposite leg of the triangle is 7.5*sin(45/2), so you can get this exactly. This is 1/2 of 1 side of the octagon, so the final answer is 16*7.5*sin(45/2) = 120sin(45/2)

sin(x/2) = sqrt((1 - cos(x))/2)
sin(45/2) = sqrt(1/2 - sqrt(2)/4)

Answer 2

Number the corners of the octagon around in a circle 1, 2, 3, ... 8.

This is easier if you label the side as s and calculate the diagonal. Then work backwards to find the side.

The longest diagonal will be from any corner to the opposite one, so let's pick 3-7.

Join 3 to 8, 1 to 6, 2 to 5. Work out the dimensions of the two triangles you've drawn in terms of s. Now work out the distance from 3 to 8. Finally, work out 7-8.

Now if this is equal to 15, you can calculate what s is.

Answer 3

Divide the octagon into eight similar triangles sharing the center of the octagon as one vertex each. The angle of each triangle touching the center vertex is 45 degrees (since there are eight simlar figures making up 360 degrees). The two sides of each triangle which start from the center are 7.5 cm each, since they're half of the longest diagonal of the octagon.

Can you figure it out from here, knowing that the remaining two angles of each triangle are similar?

Answer 4

draw a square around the octagon. you should have four triangles when done(one at each corner of the square). when you put those triangles together you get two more smaller squares. Find the area of the main square then subtract the area of the two smaller squares from the large one and that is the area of your octagon.

Answer 5

If it were a circle, the circumference would be 15pi cm. Since each side is projected from an angle of 45 (360/8), Each side would be < 15pi/8 cm. I cannot think of a way to get the exact measurement.