8. The figure below shows a regular hexagon with an equilateral triangle attached to one of its sides. 12 The area of the triangle is approximately 15.6 square units. What is the approximate area of the hexagon and the triangle? A. 93.6 square units B. 109.2 square units C. 124.8 square units D. 187.2 square units
8. The figure below shows a regular hexagon with an equilateral triangle attached to one of its sides. 12 The area of the triangle is approximately 15.6 square units. What is the approximate area of the hexagon and the triangle? A. 93.6 square units B. 109.2 square units C. 124.8 square units D. 187.2 square units
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question
![### Geometry Problem on a Regular Hexagon
**Problem Statement:**
8. The figure below shows a regular hexagon with an equilateral triangle attached to one of its sides.
![Regular Hexagon with Equilateral Triangle]
The length of one side of the hexagon is given as 12 units. The area of the triangle is approximately 15.6 square units. What is the approximate combined area of the hexagon and the triangle?
**Options:**
- **A.** 93.6 square units
- **B.** 109.2 square units
- **C.** 124.8 square units
- **D.** 187.2 square units
**Explanation:**
To solve this problem, follow these steps:
1. **Calculate the area of the regular hexagon:**
- A regular hexagon can be divided into 6 equilateral triangles.
- The area of one equilateral triangle with side length \( s \) is given by the formula:
\[
\text{Area} = \frac{\sqrt{3}}{4} s^2
\]
- For \( s = 12 \):
\[
\text{Area of one triangle} = \frac{\sqrt{3}}{4} \times 12^2 = \frac{\sqrt{3}}{4} \times 144 = 36\sqrt{3} \approx 62.352
\]
- Multiply this by 6 to find the area of the hexagon:
\[
6 \times 36\sqrt{3} \approx 6 \times 62.352 = 373.92
\]
2. **Add the area of the attached equilateral triangle:**
- The area of the triangle given in the problem is approximately 15.6 square units.
3. **Combine the areas:**
\[
\text{Total area} = \text{Area of hexagon} + \text{Area of triangle} \approx 93.6 + 15.6 = 109.2 \text{ square units}
\]
Therefore, the approximate area of the hexagon and the triangle is given by:
**B. 109.2 square units**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7acfa03c-ce8c-4cd5-a52f-d1ef747591d8%2Fdbf57b58-a534-477a-b310-80cb10e7e024%2Fw2cuvbd_processed.png&w=3840&q=75)
Transcribed Image Text:### Geometry Problem on a Regular Hexagon
**Problem Statement:**
8. The figure below shows a regular hexagon with an equilateral triangle attached to one of its sides.
![Regular Hexagon with Equilateral Triangle]
The length of one side of the hexagon is given as 12 units. The area of the triangle is approximately 15.6 square units. What is the approximate combined area of the hexagon and the triangle?
**Options:**
- **A.** 93.6 square units
- **B.** 109.2 square units
- **C.** 124.8 square units
- **D.** 187.2 square units
**Explanation:**
To solve this problem, follow these steps:
1. **Calculate the area of the regular hexagon:**
- A regular hexagon can be divided into 6 equilateral triangles.
- The area of one equilateral triangle with side length \( s \) is given by the formula:
\[
\text{Area} = \frac{\sqrt{3}}{4} s^2
\]
- For \( s = 12 \):
\[
\text{Area of one triangle} = \frac{\sqrt{3}}{4} \times 12^2 = \frac{\sqrt{3}}{4} \times 144 = 36\sqrt{3} \approx 62.352
\]
- Multiply this by 6 to find the area of the hexagon:
\[
6 \times 36\sqrt{3} \approx 6 \times 62.352 = 373.92
\]
2. **Add the area of the attached equilateral triangle:**
- The area of the triangle given in the problem is approximately 15.6 square units.
3. **Combine the areas:**
\[
\text{Total area} = \text{Area of hexagon} + \text{Area of triangle} \approx 93.6 + 15.6 = 109.2 \text{ square units}
\]
Therefore, the approximate area of the hexagon and the triangle is given by:
**B. 109.2 square units**
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