Question

an equilateral triangle has an apothem measuring 2.16 cm and a perimeter of 22.45 cm. what is the area of the equilateral triangle, rounded to the nearest tenth? 2.7 cm² 4.1 cm² 16.2 cm² 24.2 cm²

Explanation:

Step1: Recall the formula for the area of a regular polygon

The formula for the area \( A \) of a regular polygon is \( A=\frac{1}{2} \times \text{perimeter} \times \text{apothem} \). For an equilateral triangle (a regular polygon with 3 sides), we can use this formula.

Step2: Identify the given values

The perimeter \( P = 22.45 \) cm and the apothem \( a=2.16 \) cm.

Step3: Substitute the values into the formula

Substitute \( P = 22.45 \) and \( a = 2.16 \) into the formula \( A=\frac{1}{2}\times P\times a \).
\[
A=\frac{1}{2}\times22.45\times2.16
\]
First, calculate \( 22.45\times2.16 \):
\( 22.45\times2.16 = 22.45\times(2 + 0.16)=22.45\times2+22.45\times0.16 = 44.9+3.592 = 48.492 \)
Then, multiply by \( \frac{1}{2} \):
\( A=\frac{48.492}{2}=24.246 \approx 24.2 \) (rounded to the nearest tenth? Wait, no, wait. Wait, maybe I made a mistake. Wait, no, wait the options have 24.2 as an option, but let's check again. Wait, wait, the apothem of an equilateral triangle: Wait, maybe I confused the formula. Wait, no, the formula for the area of a regular polygon is \( \frac{1}{2} \times \text{perimeter} \times \text{apothem} \). Let's recalculate:

\( \frac{1}{2}\times22.45\times2.16 \). Let's compute \( 22.45\times2.16 \):

\( 22.45\times2.16 = (22 + 0.45)\times2.16=22\times2.16+0.45\times2.16 = 47.52+0.972 = 48.492 \)

Then \( \frac{48.492}{2}=24.246 \approx 24.2 \) cm². But wait, the options include 24.2. Wait, but let's check the options again. Wait, the options are 2.7, 4.1, 16.2, 24.2. So according to the calculation, the area is approximately 24.2 cm². Wait, but maybe I made a mistake in the formula? Wait, no, the formula for the area of a regular polygon is correct. Wait, but let's check the apothem of an equilateral triangle. Wait, the apothem is the distance from the center to the midpoint of a side, perpendicular to the side. For an equilateral triangle, the centroid, circumcenter, inradius (apothem) all coincide. Wait, maybe the problem is correct. So the calculation gives \( \frac{1}{2}\times22.45\times2.16 = 24.246\approx24.2 \) cm².

Wait, but let's check again. Wait, maybe I miscalculated. Let's do \( 22.45\times2.16 \):

\( 22.45\times2 = 44.9 \)

\( 22.45\times0.16 = 3.592 \)

Sum: \( 44.9 + 3.592 = 48.492 \)

Then \( 48.492\div2 = 24.246 \approx 24.2 \) cm². So the answer should be 24.2 cm². Wait, but the options have 24.2 as an option. So that's the answer.

Wait, but maybe I messed up the formula. Wait, no, the formula for the area of a regular polygon is \( A=\frac{1}{2} \times \text{perimeter} \times \text{apothem} \). So that's correct. So the area is approximately 24.2 cm².

Answer:

24.2 cm²