1. ## Using geometry...

....find the length of the side of a perfect octagon which has a measurement from its center to the perpendicular side of 8.

octagon.jpg

2. 6.6 to one decimal place?

3. The answer is correct! You might have calculated the answer using the tangent which is an acceptable method, however, without using tangent, sin, or cosin, could you still logically solve it?

4. I did indeed use tan - and there's no way I would have thought of any other way! My maths courses were a long time ago - I thought I did well thinking of the tan method!

5. ...And you sure did. Every time I use them with angles, I have to take a quick refresher course.

6. You could approximate the side without trig since the triangle formed by the 8, the bisected side, and a line from the center to the top right angle is "almost" a 30-60-90 triangle with a relationship for the sides. The side of the octagon would be (smaller than): 2*8/SQRT(3).

But, that's not what you wanted. Here's a non-trig approach.

So, think of the octagon as a square with the corners clipped off. The clipped off corners are each 45-45-90 triangles. If a leg is x, then the hypotenuse (side of the octagon) is x*SQRT(2). If the corners weren't clipped off, then the side of the square is made up of three segments and they're in the ratio 1: SQRT(2) : 1 and the side of the octagon is the longer of the three segments. Then, the fraction of the whole diameter of the octagon (in this case, 16) is the length of one of the sides of the octagon is SQRT(2) / [2+SQRT(2)]. Multiply this by 16 and you get the side of the octagon. Phew. USE TRIG!!

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