Question

A yield sign is a regular triangle. Its apothem measures 10.4 inches. What is the area of the sign to the nearest whole number?(1 point)
The area of the yield sign is approximately
in.2
.
Remaining Attempts : 2

Answer 1

To find the area of a regular triangle (in this case, the yield sign), we can use the formula that relates the area \( A \) of a regular polygon to its apothem \( a \) and its perimeter \( P \):

\[
A = \frac{1}{2} \times a \times P
\]

For a regular triangle, the perimeter \( P \) can be found if we know the length of one side \( s \). The side length \( s \) can be calculated from the apothem \( a \) using the following relationship:

\[
a = \frac{s}{\sqrt{3}}
\]

Given that the apothem \( a = 10.4 \) inches, we can solve for \( s \):

\[
10.4 = \frac{s}{\sqrt{3}} \implies s = 10.4 \sqrt{3}
\]

Now, calculating \( s \):

\[
s \approx 10.4 \times 1.732 \approx 18.04 \text{ inches}
\]

Now we can find the perimeter \( P \) of the triangle:

\[
P = 3s = 3 \times 18.04 \approx 54.12 \text{ inches}
\]

Now, we can substitute \( a \) and \( P \) back into the area formula:

\[
A = \frac{1}{2} \times 10.4 \times 54.12
\]

Calculating the area:

\[
A \approx \frac{1}{2} \times 10.4 \times 54.12 \approx 282.4512 \text{ square inches}
\]

Rounding to the nearest whole number:

\[
A \approx 282 \text{ square inches}
\]

So the area of the yield sign is approximately **282 in²**.