Question

a regular hexagon is shown.
based on the information in the diagram, what is the approximate area of this hexagon in square inches?
a. 165.6 in²
b. 193.2 in²
c. 220.8 in²
d. 331.2 in²

Explanation:

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

A regular hexagon can be divided into 6 equilateral triangles, or we can use the formula \( A = \frac{1}{2} \times perimeter \times apothem \). The apothem (the distance from the center to the midpoint of a side, forming a right angle) is given as \( 6.9 \) inches. First, we need to find the side length of the hexagon. In a regular hexagon, the side length (\( s \)) is equal to the radius of the circumscribed circle, and the apothem (\( a \)) is related to the side length by \( a = s \times \frac{\sqrt{3}}{2} \). So we can solve for \( s \): \( s=\frac{2a}{\sqrt{3}} \). But maybe it's easier to note that the perimeter (\( P \)) of a regular hexagon is \( 6s \). Alternatively, since we can also think of the hexagon as 6 isosceles triangles with base \( s \) and height (apothem) \( 6.9 \). Wait, actually, the formula \( A=\frac{1}{2} \times P \times a \) is correct, where \( P \) is the perimeter and \( a \) is the apothem. But we need to find the side length. Wait, maybe there's a mistake here—wait, in the diagram, the apothem is 6.9? Wait, no, wait the diagram shows a regular hexagon, and the apothem (the dotted line with the right angle) is 6.9? Wait, no, maybe the side length is equal to the apothem? No, in a regular hexagon, the apothem is \( \frac{\sqrt{3}}{2} s \approx 0.866s \). Wait, maybe the problem has the apothem as 6.9? Wait, no, maybe the side length is equal to the apothem? Wait, no, let's check the answer choices. Let's assume that the apothem is 6.9, and the side length \( s \) can be found from the apothem: \( a = s \times \frac{\sqrt{3}}{2} \), so \( s=\frac{2a}{\sqrt{3}} \). But maybe the problem actually has the side length equal to the apothem? Wait, no, maybe I misread. Wait, the diagram shows a regular hexagon with apothem 6.9? Wait, no, maybe the length given is the side length? Wait, no, the arrow is pointing to the apothem (the dotted line with the right angle). Wait, maybe the problem is that the apothem is 6.9, and the side length is equal to the apothem? No, that can't be. Wait, maybe the problem has a typo, or maybe I should use the formula for the area of a regular hexagon as \( A = 6 \times \frac{1}{2} \times s \times a \), where \( s \) is the side length and \( a \) is the apothem. Wait, but we need to find \( s \). Wait, alternatively, maybe the side length is equal to the apothem? No, let's calculate. Let's suppose that the apothem \( a = 6.9 \). Then, since \( a = s \times \frac{\sqrt{3}}{2} \), \( s=\frac{2 \times 6.9}{\sqrt{3}} \approx \frac{13.8}{1.732} \approx 7.97 \approx 8 \) inches. Then the perimeter \( P = 6 \times 8 = 48 \) inches. Then the area \( A=\frac{1}{2} \times 48 \times 6.9 = 24 \times 6.9 = 165.6 \). Wait, but that's option A. But wait, maybe the side length is equal to the apothem? Wait, no, if the side length is 6.9, then the apothem would be \( 6.9 \times \frac{\sqrt{3}}{2} \approx 6.9 \times 0.866 \approx 5.97 \), which is not 6.9. Wait, maybe the diagram has the apothem as 6.9, and the side length is such that the apothem is 6.9. Wait, let's re-express the formula. The area of a regular hexagon is also \( A = \frac{3\sqrt{3}}{2} s^2 \). If we use the apothem formula: \( A=\frac{1}{2} \times 6s \times a = 3s \times a \). So if \( a = 6.9 \), and we can find \( s \) from \( a = \frac{\sqrt{3}}{2} s \), so \( s = \frac{2a}{\sqrt{3}} \). Then \( A = 3 \times \frac{2a}{\sqrt{3}} \times a = \frac{6a^2}{\sqrt{3}} = 2\sqrt{3}a^2 \approx 3.464a^2 \). Then \( a = 6.9 \), so \( A \approx 3.464 \times (6.9)^2 \approx 3.464 \times 47.61 \approx 165.6 \), which is option A. Wait, but let's check the answer choices. Option A is 165.6, which matches this calculation. Alternatively, maybe the problem has the side length as 6.9? Wait, if the side length \( s = 6.9 \), then the apothem \( a = s \times \frac{\sqrt{3}}{2} \approx 6.9 \times 0.866 \approx 5.97 \). Then the area would be \( \frac{1}{2} \times 6 \times 6.9 \times 5.97 \approx 3 \times 6.9 \times 5.97 \approx 3 \times 41.193 \approx 123.58 \), which is not one of the options. So that can't be. Therefore, the apothem must be 6.9, and the side length is \( \frac{2 \times 6.9}{\sqrt{3}} \approx 8 \), perimeter 48, area \( \frac{1}{2} \times 48 \times 6.9 = 165.6 \), which is option A. Wait, but let's confirm. The formula for the area of a regular hexagon is \( A = \frac{3\sqrt{3}}{2} s^2 \) or \( A = \frac{1}{2} \times perimeter \times apothem \). Using the second formula: perimeter is 6 times the side length, apothem is the distance from center to a side (the height of each of the 6 equilateral triangles? No, the apothem is the height of the isosceles triangle with two sides as the radius and base as the side length). Wait, in a regular hexagon, the radius (distance from center to a vertex) is equal to the side length. The apothem is the distance from center to a side, which is \( \frac{\sqrt{3}}{2} s \approx 0.866s \). So if the apothem is 6.9, then \( s = \frac{6.9 \times 2}{\sqrt{3}} \approx 8 \), as before. Then perimeter is \( 6 \times 8 = 48 \). Then area is \( \frac{1}{2} \times 48 \times 6.9 = 24 \times 6.9 = 165.6 \), which is option A.

Step2: Calculate the area using the formula

Using \( A=\frac{1}{2} \times perimeter \times apothem \). First, find the side length \( s \) from the apothem \( a = 6.9 \). Since \( a = s \times \frac{\sqrt{3}}{2} \), \( s=\frac{2a}{\sqrt{3}} \approx \frac{13.8}{1.732} \approx 8 \) inches. Then the perimeter \( P = 6s = 6 \times 8 = 48 \) inches. Now, plug into the area formula: \( A=\frac{1}{2} \times 48 \times 6.9 = 24 \times 6.9 = 165.6 \) square inches.

Answer:

A. \( 165.6 \, \text{in}^2 \)