To determine the area of an equilateral triangle with a given perimeter of 24 inches, we'll follow these steps:
1. Determine the side length of the equilateral triangle:
- Since the perimeter of an equilateral triangle is the sum of the lengths of all three equal sides, we can find the side length by dividing the perimeter by 3.
[tex]\[
\text{Side length} = \frac{\text{Perimeter}}{3} = \frac{24}{3} = 8 \text{ inches}
\][/tex]
2. Use the side length to find the area:
- The formula for the area of an equilateral triangle is given by:
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} \times \text{side length}^2
\][/tex]
- Plugging in the side length we determined earlier:
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} \times 8^2
\][/tex]
- First, calculate \(8^2\):
[tex]\[
8^2 = 64
\][/tex]
- Then multiply by \(\frac{\sqrt{3}}{4}\):
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} \times 64 \approx 27.712812921102035 \text{ square inches}
\][/tex]
3. Round the area to the nearest tenth:
- The area calculated above is approximately \(27.712812921102035\) square inches.
- Rounding to the nearest tenth:
[tex]\[
27.712812921102035 \approx 27.7 \text{ square inches}
\][/tex]
Thus, the area of the equilateral triangle, rounded to the nearest tenth of square inch, is:
[tex]\[
\text{Area} = 27.7 \text{ square inches}
\][/tex]
