Question

A stop sign is a regular octagon. Each side of the sign is 12.6 in. long. The area of the stop sign is 770 in^2. What is the length of the apothem to the nearest whole number?
and.....
The logo for a school is an equilateral triangle inscribed inside a circle. The seniors are painting the logo on an outside wall of a school. The radius of the circle will be 6 feet. Find the area of the triangle.
How can I solve, could you explain me.

Answer 1

Use the Pythagorean theorem:
\[x^2 = a^2 + b^2\]a is simply 12.6/2 = 6.3
b is 6.3 + c. To find this unknown c, notice that
\[c^2 + c^2 = 12.6^2\] in the upper right hand triangle.
\[c = \sqrt{\frac{ 12.6^2 }{ 2 }}\]
In our original equation,
\[x^2 = 6.3^2 + (\sqrt{\frac{ 12.6^2 }{ 2 }})^2\]
\[x = \sqrt{6.3^2 + \frac{ 12.6^2 }{ 2 }}\]
This is the apothem, I think.

Answer 2

Ooppssss. While substituting b in the eqation, I forgot that b=6.3+c. So it should be
\[x^2 = 6.4^2 + (6.3+\sqrt{\frac{12.6^2}{2}})^2\]