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
Related questions
Question
Find the surface area of the cone, including the base.
![### Understanding the Geometry of a Right Circular Cone
In geometry and mathematics, a right circular cone is a three-dimensional shape that is easily identifiable by its circular base and a single vertex that is not in the same plane as the base. A common characteristic of a right circular cone is that the line segment from the apex (top point) of the cone to the center of the base forms a right angle with the base.
#### Components of the Cone in the Diagram:
1. **Height (h)**: The perpendicular distance from the base to the apex. In this diagram, the height of the cone is labeled as **4 cm**.
2. **Radius (r)**: The distance from the center of the base to any point on the circumference of the base. Here, it is provided as **3 cm**.
3. **Base**: A circular area where the radius is 3 cm.
The diagram of the cone provided emphasizes the height, represented by a red line segment, and the radius of the base. The black dashed line indicates the circular base, and a small black square near the base signifies the right angle formed by the height and the base's radius.
This visual representation is particularly useful for understanding how the dimensions of the cone relate to each other. By knowing the height and the radius, one can apply various geometrical formulas such as those for surface area and volume.
Using the Pythagorean theorem, one can also determine the slant height (s) of the cone which is the length of the line segment from the apex to any point on the circumference of the base. For this cone:
\[ s = \sqrt{(h^2 + r^2)} = \sqrt{(4^2 + 3^2)} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{cm} \]
Here, the slant height is **5 cm**.
### Practical Applications:
Understanding these measurements and the relationships between them is crucial for various real-world applications, including engineering, architecture, and any field that involves three-dimensional modeling.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2c5f87c-df2b-4eb5-a2d6-0f43ecec4efa%2Fb34c9632-a300-4ab8-83a4-49c5ca0c9935%2Fgr9czla_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Geometry of a Right Circular Cone
In geometry and mathematics, a right circular cone is a three-dimensional shape that is easily identifiable by its circular base and a single vertex that is not in the same plane as the base. A common characteristic of a right circular cone is that the line segment from the apex (top point) of the cone to the center of the base forms a right angle with the base.
#### Components of the Cone in the Diagram:
1. **Height (h)**: The perpendicular distance from the base to the apex. In this diagram, the height of the cone is labeled as **4 cm**.
2. **Radius (r)**: The distance from the center of the base to any point on the circumference of the base. Here, it is provided as **3 cm**.
3. **Base**: A circular area where the radius is 3 cm.
The diagram of the cone provided emphasizes the height, represented by a red line segment, and the radius of the base. The black dashed line indicates the circular base, and a small black square near the base signifies the right angle formed by the height and the base's radius.
This visual representation is particularly useful for understanding how the dimensions of the cone relate to each other. By knowing the height and the radius, one can apply various geometrical formulas such as those for surface area and volume.
Using the Pythagorean theorem, one can also determine the slant height (s) of the cone which is the length of the line segment from the apex to any point on the circumference of the base. For this cone:
\[ s = \sqrt{(h^2 + r^2)} = \sqrt{(4^2 + 3^2)} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{cm} \]
Here, the slant height is **5 cm**.
### Practical Applications:
Understanding these measurements and the relationships between them is crucial for various real-world applications, including engineering, architecture, and any field that involves three-dimensional modeling.
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