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
Find the surface area and Volume of the regular pyramids.
![### Surface Area and Volume of Regular Pyramids
To find the surface area and the volume for the following regular pyramids, examine the given data for each pyramid and use appropriate mathematical formulas.
#### Pyramid 10
- **Dimensions**:
- Slant Height: 16 units
- Apothem: 4 units
#### Pyramid 11
- **Dimensions**:
- Height: 27 units
- Apothem: 9 units
#### Pyramid 12
- **Dimensions**:
- Lateral Edge: 8 units
- Diagonal: 4 units
### Explanation of Diagrams
- **Diagram 10**:
This pyramid is represented with its slant height and apothem indicated. The slant height is the distance from the apex to the midpoint of one of the base edges, while the apothem is the distance from the center of the base to the midpoint of one of the base edges.
- **Diagram 11**:
For this pyramid, the height and apothem are provided. The height is the perpendicular distance from the apex to the center of the base, and the apothem is the distance from the center of the base to the middle point of a side of the base.
- **Diagram 12**:
The dimensions include the lateral edge and diagonal. The lateral edge is the distance from the apex to a vertex of the base, and the diagonal is referring to the distance in the base.
### Calculations
To find the **surface area** of a regular pyramid, use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
The **lateral area** can be found using:
\[ \text{Lateral Area} = \frac{1}{2} \times \text{Perimeter of the base} \times \text{Slant Height} \]
To calculate the **volume** of a regular pyramid, the formula is:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
### Example:
For Pyramid 11:
- The base area can be calculated if the shape of the base is known. If it is a square base, for example, you can find the side length using the given apothem and some trigonometry or geometry.
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Transcribed Image Text:### Surface Area and Volume of Regular Pyramids
To find the surface area and the volume for the following regular pyramids, examine the given data for each pyramid and use appropriate mathematical formulas.
#### Pyramid 10
- **Dimensions**:
- Slant Height: 16 units
- Apothem: 4 units
#### Pyramid 11
- **Dimensions**:
- Height: 27 units
- Apothem: 9 units
#### Pyramid 12
- **Dimensions**:
- Lateral Edge: 8 units
- Diagonal: 4 units
### Explanation of Diagrams
- **Diagram 10**:
This pyramid is represented with its slant height and apothem indicated. The slant height is the distance from the apex to the midpoint of one of the base edges, while the apothem is the distance from the center of the base to the midpoint of one of the base edges.
- **Diagram 11**:
For this pyramid, the height and apothem are provided. The height is the perpendicular distance from the apex to the center of the base, and the apothem is the distance from the center of the base to the middle point of a side of the base.
- **Diagram 12**:
The dimensions include the lateral edge and diagonal. The lateral edge is the distance from the apex to a vertex of the base, and the diagonal is referring to the distance in the base.
### Calculations
To find the **surface area** of a regular pyramid, use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
The **lateral area** can be found using:
\[ \text{Lateral Area} = \frac{1}{2} \times \text{Perimeter of the base} \times \text{Slant Height} \]
To calculate the **volume** of a regular pyramid, the formula is:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
### Example:
For Pyramid 11:
- The base area can be calculated if the shape of the base is known. If it is a square base, for example, you can find the side length using the given apothem and some trigonometry or geometry.
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