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| Math Central | Quandaries & Queries |
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Question from CHARLES: I have a 20" octagon and it has 8 parts. In inches how big is each part. I was told each part was 5.86".The top of the octagon breakes into 3 equal parts dont equal 20".When i did my math i thought it should be 6 5/8" which is closer to 20" |
Hi Charles.
By "twenty inch circle", I assume you mean the diameter is 20 inches. Therefore, the octagon's opposite corners are 20 inches apart. If you draw (on paper or in your mind's eye) an octagon and the middle point of it, joining the corners to the center, you will see eight equivalent triangular segments. The lengths of the sides of the triangles are 10 inches, because they equal the radius of the circle. However the length of the side close to the perimeter of the circle is the length you need to find.
Fortunately, there is a formula for the length of the third side of an isosceles triangle if you know the size of the angle joining the common sides:
x = 2 r sin (A/2)
where r is the length of the paired sides, A is the angle joining them, and x is the length of the opposite side.
Remember that a full circle is 360 degrees. This means each triangular segment of an octagon has a 360 / 8 = 45 degree angle joining the equal sides.
Can you finish the problem from here? Use your calculator to get the final answer.
Cheers,
Stephen La Rocque.
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