If an octagon has a circumference of 24 what is its diameter?

Answer 1

I assume that when you say "circumference", you really mean "perimeter" of the octagon. I also assume that by saying, "diameter", you really refer to the widest distance between two points on the octagon. An octagon has 8 sides, so if you divide 24 by 8, then you have 3 units of length per side. Then if you were to draw lines from the center of the octagon out to the ends of the segments of the perimeter, then those lines would be 45 degrees apart from each other. In order to find the "diameter", you need to find the length of one of those lines drawn from the center of the octagon, then double it. To do this, you need to perform some trigonometry. To perform the trig, the easiest thing would be to use a right triangle. By drawing a line from the center of the octagon, perpendicular (at a right angle) to one of the outside segments of the octagon, you bisect the 45 degree angle created by the two lines on either side of it. So then you have created a right triangle inside of the octagon, with a 1.5 unit long side, and an opposing 22.5 degree angle. In this case, you want to find out the length of the longest side of this triangle, which is the hypotenuse. So, the sine of the angle 22.5 degrees is equal to 1.5 units divided by the length of the hypotenuse. If you solve for the hypotenuse, you get 3.92 units, rounded. Double this gives you 7.84 units across the octagon.