Question

if an octagon is inscribed in a circle with a radius of 10cm. what will be the side of the octagon?

Answer 1

Since our octagon can be split into 8 triangles around the circle, each vertex angle of these triangles can be found to be \(\frac{2\pi}8=\frac\pi4\). Using the fact that our triangles have two congruent sides (radii for the circumscribed circle, therefore of length \(10\)), we can use the law of sines to determine our unknown side length \(L\): \(\frac{L}{\sin\frac\pi4}=\frac{10}{\sin\frac\pi8}\)$$\frac{L}{\sin\frac\pi4}=\frac{10}{\sin\frac\pi8}\\L=10\frac{\sin\frac\pi4}{\sin\frac\pi8}=5\sqrt2\csc\frac\pi8$$