Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter9: Surfaces And Solids
Section9.2: Pyramids, Area, And Volume
Problem 35E
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Question
![### Problem Statement
4. Determine the volume of a pyramid that has a 6-cm-by-6-cm square base and four faces that are equilateral triangles. Explain your reasoning.
### Detailed Explanation
To determine the volume of a pyramid with a square base and triangular faces, we need to follow these steps:
1. **Determine the base area (A):**
The base of the pyramid is a square with each side measuring 6 cm.
The area of a square is given by \(A = a^2\), where \(a\) is the side length.
\[
A = 6 \, \text{cm} \times 6 \, \text{cm} = 36 \, \text{cm}^2
\]
2. **Determine the height (h) of the pyramid:**
Given that the four faces are equilateral triangles, we can find the height of the pyramid using properties of equilateral triangles and some trigonometry.
Each face is an equilateral triangle with side length \(6 \, \text{cm}\). The height of an equilateral triangle (h_t) can be determined by:
\[
h_t = \frac{\sqrt{3}}{2} \times \text{side length} = \frac{\sqrt{3}}{2} \times 6 \, \text{cm} = 3\sqrt{3} \, \text{cm}
\]
This height forms the slant height of the pyramid.
For an equilateral triangle, we can drop a perpendicular from the apex of the pyramid to the center of its base, which is also the center of the square base. This perpendicular height forms a right triangle with the base edge of the square to its center.
3. **Determine the volume (V) of the pyramid:**
The volume of a pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
But first, we need the vertical height \(h\) of the pyramid. The height \(h\) can be found by using the Pythagorean theorem in the right triangle formed by the pyramid's height, the part of the diagonal of the square (half of diagonal) and the slant height.
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Transcribed Image Text:### Problem Statement
4. Determine the volume of a pyramid that has a 6-cm-by-6-cm square base and four faces that are equilateral triangles. Explain your reasoning.
### Detailed Explanation
To determine the volume of a pyramid with a square base and triangular faces, we need to follow these steps:
1. **Determine the base area (A):**
The base of the pyramid is a square with each side measuring 6 cm.
The area of a square is given by \(A = a^2\), where \(a\) is the side length.
\[
A = 6 \, \text{cm} \times 6 \, \text{cm} = 36 \, \text{cm}^2
\]
2. **Determine the height (h) of the pyramid:**
Given that the four faces are equilateral triangles, we can find the height of the pyramid using properties of equilateral triangles and some trigonometry.
Each face is an equilateral triangle with side length \(6 \, \text{cm}\). The height of an equilateral triangle (h_t) can be determined by:
\[
h_t = \frac{\sqrt{3}}{2} \times \text{side length} = \frac{\sqrt{3}}{2} \times 6 \, \text{cm} = 3\sqrt{3} \, \text{cm}
\]
This height forms the slant height of the pyramid.
For an equilateral triangle, we can drop a perpendicular from the apex of the pyramid to the center of its base, which is also the center of the square base. This perpendicular height forms a right triangle with the base edge of the square to its center.
3. **Determine the volume (V) of the pyramid:**
The volume of a pyramid is given by:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
But first, we need the vertical height \(h\) of the pyramid. The height \(h\) can be found by using the Pythagorean theorem in the right triangle formed by the pyramid's height, the part of the diagonal of the square (half of diagonal) and the slant height.
Di
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