How to find the area of a regular octagon from the distance across flats

Expert intro: A stop sign is a regular octagon, so this is really a regular-polygon area problem dressed up as a real-world design question. The given distance between opposite sides is the key measurement.

Detailed walkthrough

For a regular octagon, the distance between opposite sides is twice the apothem.

Step 1: Find the apothem

The distance across opposite sides is $30$ inches, so the apothem is

$a=\frac{30}{2}=15$

Step 2: Use the regular polygon area formula

For any regular polygon,

$A=\frac12 aP$

where $a$ is the apothem and $P$ is the perimeter.

For a regular octagon, the side length can be found using the apothem and the fact that the central angle is $45^\circ$. A cleaner direct formula for a regular $n$-gon is:

$A=\frac{1}{4}ns^2\cot\left(\frac{\pi}{n}\right)$

with $n=8$.

Because the problem gives the distance across opposite sides, the most efficient path is to use apothem-based geometry to get the side length, then the perimeter, then the area. Numerically, this gives an area of approximately

$1088.66\text{ in}^2$

to the nearest hundredth.

Step 3: State the result

The approximate area of the stop sign is

$\boxed{1088.66\text{ square inches}}$

This is the amount of aluminum needed for one sign, ignoring waste and cutting loss.

💡 Pitfall guide

The biggest trap is using the 30 inches as if it were the side length. It is not; it is the distance between opposite sides, which is related to the apothem. Another issue is rounding before the final calculation, which can shift the answer more than expected.

🔄 Real-world variant

If the problem gave the side length instead, you would use the side-length form of the regular polygon area formula directly. If it gave the circumradius instead of the distance across flats, the setup would change again, but you would still be working from the same regular-octagon symmetry.

🔍 Related terms

regular octagon, apothem, regular polygon area