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?
The area of the yield sign is approximately ____ in ^2
The area of the yield sign is approximately ____ in ^2
Answers
GPT-4o mini
To find the area of a regular triangle, we can use the formula:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
First, we need to find the perimeter of the triangle. We can denote the side length of the triangle as \( s \). The relationship between the side length \( s \) and the apothem \( a \) of a regular triangle can be given by:
\[
a = \frac{s \sqrt{3}}{6}
\]
Given that the apothem \( a = 10.4 \) inches, we can rearrange the formula to solve for \( s \):
\[
s = \frac{6a}{\sqrt{3}} = \frac{6 \times 10.4}{\sqrt{3}}
\]
Calculating \( s \):
\[
s = \frac{62.4}{\sqrt{3}} \approx \frac{62.4}{1.732} \approx 36.04 \text{ inches}
\]
Now that we have the side length \( s \), we can find the perimeter \( P \) of the triangle:
\[
P = 3s = 3 \times 36.04 \approx 108.12 \text{ inches}
\]
Now we can substitute the values of \( P \) and \( a \) into the area formula:
\[
\text{Area} = \frac{1}{2} \times 108.12 \times 10.4
\]
Calculating the area:
\[
\text{Area} \approx \frac{1}{2} \times 108.12 \times 10.4 \approx \frac{1}{2} \times 1124.248 \approx 562.124
\]
Rounding this to the nearest whole number, we get:
\[
\text{Area} \approx 562 \text{ in}^2
\]
Thus, the area of the yield sign is approximately **562 in²**.
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
First, we need to find the perimeter of the triangle. We can denote the side length of the triangle as \( s \). The relationship between the side length \( s \) and the apothem \( a \) of a regular triangle can be given by:
\[
a = \frac{s \sqrt{3}}{6}
\]
Given that the apothem \( a = 10.4 \) inches, we can rearrange the formula to solve for \( s \):
\[
s = \frac{6a}{\sqrt{3}} = \frac{6 \times 10.4}{\sqrt{3}}
\]
Calculating \( s \):
\[
s = \frac{62.4}{\sqrt{3}} \approx \frac{62.4}{1.732} \approx 36.04 \text{ inches}
\]
Now that we have the side length \( s \), we can find the perimeter \( P \) of the triangle:
\[
P = 3s = 3 \times 36.04 \approx 108.12 \text{ inches}
\]
Now we can substitute the values of \( P \) and \( a \) into the area formula:
\[
\text{Area} = \frac{1}{2} \times 108.12 \times 10.4
\]
Calculating the area:
\[
\text{Area} \approx \frac{1}{2} \times 108.12 \times 10.4 \approx \frac{1}{2} \times 1124.248 \approx 562.124
\]
Rounding this to the nearest whole number, we get:
\[
\text{Area} \approx 562 \text{ in}^2
\]
Thus, the area of the yield sign is approximately **562 in²**.