Find the length of one side, the perimeter and area of a regular octagon given the distance between two opposite sides (span).
Solution to the Problem:
The octagon has 8 interior angles. The measure of one interior angle ABD is given by
(8 - 2)*180 / 8 = 135 degrees
Draw AC and BC so that they are perpendicular at C. Hence the measure of ABC is given by
135 - 90 = 45 degrees
Note that triangle ABC is right and isosceles. We now write that the given distance of 10 cm is the sum of 2y and x
2y + x = 10
Using Pythagora's theorem, we can also write
2y2 = x2
We now solve the above system of equations to obtain x, the length of one side.
x = 10 / (1 + sqrt(2)) cm
The perimeter P is given by.
P = 8*10 / (1 + sqrt(2)) = 80 / (1 + sqrt(2)) cm
The area A may be calculated by subtracting the areas of the 4 right triangles from the area of the large square of side 10 cm.
A = 10*10 - 4(1/2)(y2) = 100 - 2(y2) = 100 - (x2) = 200(sqrt(2)-1) cm2
More references on geometry.
Geometry Tutorials, Problems and Interactive Applets.