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
Concept explainers
Cylinders
A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.
Cones
A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.
Question
Find the lateral surface area of the cone.
![Certainly, here is the transcription suitable for an Educational website:
---
### Understanding the Geometry of a Right Circular Cone
The diagram illustrates a right circular cone with two important measurements:
1. **Radius (r) of the base**: The radius of the base of the cone is given as 4 inches. This is the distance from the center of the base to any point on its perimeter.
2. **Height (h) of the cone**: The height of the cone is 6 inches. This is the perpendicular distance from the base to the apex (or vertex) of the cone.
**Diagram Explanation**:
- The shape shown is a three-dimensional geometric figure called a "cone."
- The base of the cone, shown with a dashed line, is a circle.
- The vertical line extending from the center of the circular base to the apex is the height of the cone.
- The slant height (the distance from any point on the perimeter of the base to the apex) is not provided but is an important measure in cone geometry.
This geometric figure is commonly studied in math and engineering courses, specifically when exploring topics in solid geometry, surface area, and volume calculations.
### Key Formulas:
1. **Volume of a Cone**:
\[ V = \frac{1}{3} \pi r^2 h \]
2. **Surface Area of a Cone**:
\[ A = \pi r (r + l) \]
Here, \( l \) is the slant height, which can be determined using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
By substituting the given values:
- Radius (\( r \)) = 4 inches
- Height (\( h \)) = 6 inches
You can calculate specific properties such as the volume and surface area of this cone.
---
This educational content provides an in-depth explanation of the cone's geometry and guides students through understanding and applying mathematical formulas.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2c5f87c-df2b-4eb5-a2d6-0f43ecec4efa%2F444ebc0c-4505-45d1-a6ad-7d0637c14521%2F1zcqaec_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly, here is the transcription suitable for an Educational website:
---
### Understanding the Geometry of a Right Circular Cone
The diagram illustrates a right circular cone with two important measurements:
1. **Radius (r) of the base**: The radius of the base of the cone is given as 4 inches. This is the distance from the center of the base to any point on its perimeter.
2. **Height (h) of the cone**: The height of the cone is 6 inches. This is the perpendicular distance from the base to the apex (or vertex) of the cone.
**Diagram Explanation**:
- The shape shown is a three-dimensional geometric figure called a "cone."
- The base of the cone, shown with a dashed line, is a circle.
- The vertical line extending from the center of the circular base to the apex is the height of the cone.
- The slant height (the distance from any point on the perimeter of the base to the apex) is not provided but is an important measure in cone geometry.
This geometric figure is commonly studied in math and engineering courses, specifically when exploring topics in solid geometry, surface area, and volume calculations.
### Key Formulas:
1. **Volume of a Cone**:
\[ V = \frac{1}{3} \pi r^2 h \]
2. **Surface Area of a Cone**:
\[ A = \pi r (r + l) \]
Here, \( l \) is the slant height, which can be determined using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
By substituting the given values:
- Radius (\( r \)) = 4 inches
- Height (\( h \)) = 6 inches
You can calculate specific properties such as the volume and surface area of this cone.
---
This educational content provides an in-depth explanation of the cone's geometry and guides students through understanding and applying mathematical formulas.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning