# Question #bcedf

Sep 28, 2017

Length of each side is 248.53 mm 2.d.p.
Interior angle is ${135}^{o}$

#### Explanation:

I don't know if I will be able to explain this with words alone. This is where you desperately need geometrical diagrams. I'll try my best.

If you construct isosceles triangles on the inside of the octagon the angle at there apex will be:

$\frac{{360}^{o}}{8}$ (8 being the number of sides.)

This gives $\left({45}^{o}\right)$ Take a perpendicular bisector of this angle. This will then bisect the angle and form a right angle with the base (giving you half the length of one of the sides of the octagon). The angle will then be:
$\frac{{45}^{o}}{2} = \left({22.5}^{o}\right)$
We know the length of one side of this triangle, because it will be the length of the radius, which is $\frac{600 m m}{2} = 300$mm.
We now have a right angled triangle with angles ${90}^{o}$ and ${22.5}^{o}$. The third angle will be half of one of the interior angles of the octagon.
Sum of interior angles is:

$180 n - 360$

Where $n$ is number of sides.

So: ${180}^{o} \left(8\right) - {360}^{o} = {1080}^{o}$

Each angle is then: $\frac{{1080}^{o}}{8} = {135}^{o}$

Half this angle to get the third angle of the triangle:

$\frac{{135}^{o}}{2} = {67.5}^{o}$

We now have a right angled triangle with angles ${22.5}^{o} , {67.5}^{o} \mathmr{and} {90}^{o}$. with one side of $300$( this is the radius).

If we label the sides of the triangle a b c ( c being the hypotenuse ) and opposite angles A B C respectively.

We need to find the length of side a.

Using the Sine Rule $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{S \in C}{c}$

$\sin \frac{22.5}{a} = \sin \frac{67.5}{300}$

Rearranged gives:

$a = \frac{300 \sin \left(22.5\right)}{\sin \left(67.5\right)} \implies a = 124.2641$

4 .d.p.

This is HALF the length of one of the sides of the octagon, so:

$124.2641 \times 2 = 248.53$

2 .d.p.

So we have:

Length of octagon side 248.53mm.

Interior angle ${135}^{o}$

Hope this helps you. It would have been much easier for me and for you if we could have used diagrams.