Answer :
To determine which formula finds the surface area of a right cone, let's analyze each one step-by-step:
1. Understanding the Components:
- The surface area [tex]\(A\)[/tex] of a right cone includes two parts: the base area [tex]\(A_{\text{base}}\)[/tex] and the lateral area [tex]\(A_{\text{lateral}}\)[/tex].
- The base area [tex]\(A_{\text{base}}\)[/tex] is given by:
[tex]\[ A_{\text{base}} = \pi r^2 \][/tex]
- The lateral area [tex]\(A_{\text{lateral}}\)[/tex] is given by:
[tex]\[ A_{\text{lateral}} = \pi r s \][/tex]
- Therefore, the total surface area [tex]\(A\)[/tex] is:
[tex]\[ A = A_{\text{base}} + A_{\text{lateral}} = \pi r^2 + \pi r s \][/tex]
2. Analyzing Each Option:
- Option A: [tex]\(A_{\text{base}} + 2\pi r^2\)[/tex]
- This formula incorrectly adds an extra term. The correct base area should only be [tex]\(\pi r^2\)[/tex], not [tex]\(2\pi r^2\)[/tex].
- Incorrect.
- Option B: [tex]\(A_{\text{base}} + A_{\text{lateral}}\)[/tex]
- This formula correctly adds the base area and the lateral area, which follows the standard surface area formula.
- Correct.
- Option C: [tex]\(2\pi r^2 + 2\pi r h\)[/tex]
- This formula mixes the terms for the surface area of a cylinder, not a cone.
- Incorrect.
- Option D: [tex]\(2A_{\text{lateral}} + \pi r^2\)[/tex]
- This formula incorrectly doubles the lateral area and improperly adds an extra term.
- Incorrect.
- Option E: [tex]\(\pi r s\)[/tex]
- This formula only represents the lateral area and does not include the base area.
- Incorrect.
Given the above analysis, the correct formula for finding the surface area of a right cone is option:
B. [tex]\(A_{\text{base}} + A_{\text{lateral}}\)[/tex]
Thus, the correct formula to find the surface area of a right cone is option B, which corresponds to the choice given by the analysis.
1. Understanding the Components:
- The surface area [tex]\(A\)[/tex] of a right cone includes two parts: the base area [tex]\(A_{\text{base}}\)[/tex] and the lateral area [tex]\(A_{\text{lateral}}\)[/tex].
- The base area [tex]\(A_{\text{base}}\)[/tex] is given by:
[tex]\[ A_{\text{base}} = \pi r^2 \][/tex]
- The lateral area [tex]\(A_{\text{lateral}}\)[/tex] is given by:
[tex]\[ A_{\text{lateral}} = \pi r s \][/tex]
- Therefore, the total surface area [tex]\(A\)[/tex] is:
[tex]\[ A = A_{\text{base}} + A_{\text{lateral}} = \pi r^2 + \pi r s \][/tex]
2. Analyzing Each Option:
- Option A: [tex]\(A_{\text{base}} + 2\pi r^2\)[/tex]
- This formula incorrectly adds an extra term. The correct base area should only be [tex]\(\pi r^2\)[/tex], not [tex]\(2\pi r^2\)[/tex].
- Incorrect.
- Option B: [tex]\(A_{\text{base}} + A_{\text{lateral}}\)[/tex]
- This formula correctly adds the base area and the lateral area, which follows the standard surface area formula.
- Correct.
- Option C: [tex]\(2\pi r^2 + 2\pi r h\)[/tex]
- This formula mixes the terms for the surface area of a cylinder, not a cone.
- Incorrect.
- Option D: [tex]\(2A_{\text{lateral}} + \pi r^2\)[/tex]
- This formula incorrectly doubles the lateral area and improperly adds an extra term.
- Incorrect.
- Option E: [tex]\(\pi r s\)[/tex]
- This formula only represents the lateral area and does not include the base area.
- Incorrect.
Given the above analysis, the correct formula for finding the surface area of a right cone is option:
B. [tex]\(A_{\text{base}} + A_{\text{lateral}}\)[/tex]
Thus, the correct formula to find the surface area of a right cone is option B, which corresponds to the choice given by the analysis.
