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
.

Answer 1

To find the area of a regular triangle (like a yield sign), we can use the formula for the area \( A \) in terms of the apothem \( a \) and the perimeter \( P \):

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

However, we first need to find the perimeter \( P \). For a regular triangle, we can find the side length \( s \) using the apothem. The relationship between the side length and the apothem is given by:

\[
a = \frac{s \sqrt{3}}{6}
\]

Rearranging this formula allows us to find \( s \):

\[
s = \frac{6a}{\sqrt{3}} = 2\sqrt{3}a
\]

Substituting \( a = 10.4 \) inches:

\[
s = 2\sqrt{3}(10.4) \approx 2(1.732)(10.4) \approx 35.998 \text{ inches}
\]

Now, the perimeter \( P \) of the triangle, which has three sides, will be:

\[
P = 3s \approx 3(35.998) \approx 107.994 \text{ inches}
\]

Now we can substitute everything back into the area formula:

\[
A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 107.994 \times 10.4
\]

Calculating this:

\[
A \approx \frac{1}{2} \times 107.994 \times 10.4 \approx \frac{1}{2} \times 1123.91336 \approx 561.95668
\]

Rounding to the nearest whole number, the area of the yield sign is approximately:

\[
\boxed{562} \text{ square inches}
\]