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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![### Problem Statement:
**17. Find the total area (surface area) of a regular icosahedron if each edge has a length of 4.2 cm.**
### Explanation:
To solve this problem, students will need to use the formula for the surface area of a regular icosahedron. An icosahedron is a polyhedron with 20 equilateral triangular faces, 30 edges, and 12 vertices.
The formula for the surface area (SA) of a regular icosahedron with edge length \(a\) is:
\[ SA = 5 \sqrt{3} a^2 \]
Given:
\[ a = 4.2 \, \text{cm} \]
Substitute the given edge length into the formula to find the total surface area.
### Steps to Solve:
1. **Substitute** the given edge length into the formula:
\[
SA = 5 \sqrt{3} (4.2 \, \text{cm})^2
\]
2. **Calculate** the square of the edge length:
\[
(4.2 \, \text{cm})^2 = 17.64 \, \text{cm}^2
\]
3. **Multiply** by \(5 \sqrt{3} \):
\[
SA = 5 \sqrt{3} \times 17.64 \, \text{cm}^2
\]
4. **Simplify** the expression (using \(\sqrt{3} \approx 1.732\)):
\[
SA \approx 5 \times 1.732 \times 17.64 \, \text{cm}^2
\]
5. **Complete the multiplications**:
\[
SA \approx 5 \times 30.56 \, \text{cm}^2
\]
\[
SA \approx 152.8 \, \text{cm}^2
\]
Therefore, the total surface area of the icosahedron is approximately \(152.8 \, \text{cm}^2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55a87db1-34dd-4361-bb49-0fd4dde658a5%2Fd68d81a1-cd47-4e90-9ca0-228a4c4fc327%2Fh7bnv5w.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**17. Find the total area (surface area) of a regular icosahedron if each edge has a length of 4.2 cm.**
### Explanation:
To solve this problem, students will need to use the formula for the surface area of a regular icosahedron. An icosahedron is a polyhedron with 20 equilateral triangular faces, 30 edges, and 12 vertices.
The formula for the surface area (SA) of a regular icosahedron with edge length \(a\) is:
\[ SA = 5 \sqrt{3} a^2 \]
Given:
\[ a = 4.2 \, \text{cm} \]
Substitute the given edge length into the formula to find the total surface area.
### Steps to Solve:
1. **Substitute** the given edge length into the formula:
\[
SA = 5 \sqrt{3} (4.2 \, \text{cm})^2
\]
2. **Calculate** the square of the edge length:
\[
(4.2 \, \text{cm})^2 = 17.64 \, \text{cm}^2
\]
3. **Multiply** by \(5 \sqrt{3} \):
\[
SA = 5 \sqrt{3} \times 17.64 \, \text{cm}^2
\]
4. **Simplify** the expression (using \(\sqrt{3} \approx 1.732\)):
\[
SA \approx 5 \times 1.732 \times 17.64 \, \text{cm}^2
\]
5. **Complete the multiplications**:
\[
SA \approx 5 \times 30.56 \, \text{cm}^2
\]
\[
SA \approx 152.8 \, \text{cm}^2
\]
Therefore, the total surface area of the icosahedron is approximately \(152.8 \, \text{cm}^2\).
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