Answer to Question #201022 in Geometry for Mohammed Rihan Bab

Question

A city pays a manufacturing company to produce more stop signs for the city due to an increase in traffic incidents at intersections. After doing some research, the company manager reads that a standard stop sign is a regular octagon that is 30 inches wide and 30 inches tall.

Image courtesy of Pixabay user FotoshopTofs at https://pixabay.com/en/stop-road-panel-ban-traffic-1327126/  

The manager must calculate the area of a stop sign so that he can order enough metal and paint to get the job done. What is the area of one standard-sized stop sign?

Answer 1

Solution:-

let a corner of octagon BCA , A as out to make it a triangle right angled at A

let the side of Regular octagon = x inch

each interior angle $=\frac{(8-2)(180)}{8}\\ =135 ^o\\$

$\angle ABC = 180- 135 = 45^o$

$\angle ACB= 180-135=45^o \\ \therefore ACB \ is \ Right \ triangle \ is \ isosceless \ triangle$

hence AB=AC

$\therefore \ BC^2=AB^2 \\ BC^2=AB^2 + AC^2\\ x^2=2AB^2\\ AB= \frac{x}{\sqrt{2}}$

$\therefore 30= \frac{x}{\sqrt{2}}+x+ \frac{x}{\sqrt{2}}$

$x\approx12.5 \ inches$

perimeter = 8x=$8\times 12.5 \approx100 \ inches$

area = $\frac{1}{2}\times(12.5)\times(100)\approx\boxed{625 \ inch^2}$