### Square Pyramid Calculation #### Problem Description You are given a square pyramid with the following dimensions: - Base length (\(b\)) = 6 inches - Slant height (\(s\)) = 12 inches The goal is to solve for the surface area and volume of the square pyramid. #### Diagram Explanation The provided diagram is a 3D representation of a square pyramid: - The base (\(b\)) is a square with highlighted edges. - The slant height (\(s\)) is the length from the top vertex perpendicular to the midpoint of an edge of the square base. - The pyramid's height (\(h\)) is not explicitly given but can be calculated if needed. #### Steps to Solve 1. **Surface Area Calculation:** The formula for the surface area (\(SA\)) of a square pyramid is: \[ SA = b^2 + 2b \cdot s \] where: - \(b^2\) is the area of the square base. - \(2b \cdot s\) is the combined area of the four triangular faces. 2. **Volume Calculation:** The formula for the volume (\(V\)) of a square pyramid is: \[ V = \frac{1}{3} b^2 \cdot h \] To find \(h\) (the height of the pyramid), use the Pythagorean Theorem in the right triangle formed by the height (\(h\)), half of the base length (\(\frac{b}{2}\)), and the slant height (\(s\)): \[ s^2 = h^2 + \left(\frac{b}{2}\right)^2 \] To solve for \(h\): \[ h = \sqrt{s^2 - \left(\frac{b}{2}\right)^2} \] Once you have the values, you can plug them into their respective formulas to find the surface area and volume. #### Input Fields: - **Surface Area**: - A text box where the calculated surface area will be entered (in square inches). - **Volume**: - A text box where the calculated volume will be entered (in cubic inches). #### Help Options: - **Question Help**: - Video: A video
### Square Pyramid Calculation #### Problem Description You are given a square pyramid with the following dimensions: - Base length (\(b\)) = 6 inches - Slant height (\(s\)) = 12 inches The goal is to solve for the surface area and volume of the square pyramid. #### Diagram Explanation The provided diagram is a 3D representation of a square pyramid: - The base (\(b\)) is a square with highlighted edges. - The slant height (\(s\)) is the length from the top vertex perpendicular to the midpoint of an edge of the square base. - The pyramid's height (\(h\)) is not explicitly given but can be calculated if needed. #### Steps to Solve 1. **Surface Area Calculation:** The formula for the surface area (\(SA\)) of a square pyramid is: \[ SA = b^2 + 2b \cdot s \] where: - \(b^2\) is the area of the square base. - \(2b \cdot s\) is the combined area of the four triangular faces. 2. **Volume Calculation:** The formula for the volume (\(V\)) of a square pyramid is: \[ V = \frac{1}{3} b^2 \cdot h \] To find \(h\) (the height of the pyramid), use the Pythagorean Theorem in the right triangle formed by the height (\(h\)), half of the base length (\(\frac{b}{2}\)), and the slant height (\(s\)): \[ s^2 = h^2 + \left(\frac{b}{2}\right)^2 \] To solve for \(h\): \[ h = \sqrt{s^2 - \left(\frac{b}{2}\right)^2} \] Once you have the values, you can plug them into their respective formulas to find the surface area and volume. #### Input Fields: - **Surface Area**: - A text box where the calculated surface area will be entered (in square inches). - **Volume**: - A text box where the calculated volume will be entered (in cubic inches). #### Help Options: - **Question Help**: - Video: A video
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Solve the surface area and volume of the square pyramid
b=6in
s=12 in
![### Square Pyramid Calculation
#### Problem Description
You are given a square pyramid with the following dimensions:
- Base length (\(b\)) = 6 inches
- Slant height (\(s\)) = 12 inches
The goal is to solve for the surface area and volume of the square pyramid.
#### Diagram Explanation
The provided diagram is a 3D representation of a square pyramid:
- The base (\(b\)) is a square with highlighted edges.
- The slant height (\(s\)) is the length from the top vertex perpendicular to the midpoint of an edge of the square base.
- The pyramid's height (\(h\)) is not explicitly given but can be calculated if needed.
#### Steps to Solve
1. **Surface Area Calculation:**
The formula for the surface area (\(SA\)) of a square pyramid is:
\[
SA = b^2 + 2b \cdot s
\]
where:
- \(b^2\) is the area of the square base.
- \(2b \cdot s\) is the combined area of the four triangular faces.
2. **Volume Calculation:**
The formula for the volume (\(V\)) of a square pyramid is:
\[
V = \frac{1}{3} b^2 \cdot h
\]
To find \(h\) (the height of the pyramid), use the Pythagorean Theorem in the right triangle formed by the height (\(h\)), half of the base length (\(\frac{b}{2}\)), and the slant height (\(s\)):
\[
s^2 = h^2 + \left(\frac{b}{2}\right)^2
\]
To solve for \(h\):
\[
h = \sqrt{s^2 - \left(\frac{b}{2}\right)^2}
\]
Once you have the values, you can plug them into their respective formulas to find the surface area and volume.
#### Input Fields:
- **Surface Area**:
- A text box where the calculated surface area will be entered (in square inches).
- **Volume**:
- A text box where the calculated volume will be entered (in cubic inches).
#### Help Options:
- **Question Help**:
- Video: A video](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b14f8be-e912-4da1-8445-57a3ff28b92f%2F4d122102-89ed-4ef7-8992-ac7a1fea42fe%2Fp3i888q.jpeg&w=3840&q=75)
Transcribed Image Text:### Square Pyramid Calculation
#### Problem Description
You are given a square pyramid with the following dimensions:
- Base length (\(b\)) = 6 inches
- Slant height (\(s\)) = 12 inches
The goal is to solve for the surface area and volume of the square pyramid.
#### Diagram Explanation
The provided diagram is a 3D representation of a square pyramid:
- The base (\(b\)) is a square with highlighted edges.
- The slant height (\(s\)) is the length from the top vertex perpendicular to the midpoint of an edge of the square base.
- The pyramid's height (\(h\)) is not explicitly given but can be calculated if needed.
#### Steps to Solve
1. **Surface Area Calculation:**
The formula for the surface area (\(SA\)) of a square pyramid is:
\[
SA = b^2 + 2b \cdot s
\]
where:
- \(b^2\) is the area of the square base.
- \(2b \cdot s\) is the combined area of the four triangular faces.
2. **Volume Calculation:**
The formula for the volume (\(V\)) of a square pyramid is:
\[
V = \frac{1}{3} b^2 \cdot h
\]
To find \(h\) (the height of the pyramid), use the Pythagorean Theorem in the right triangle formed by the height (\(h\)), half of the base length (\(\frac{b}{2}\)), and the slant height (\(s\)):
\[
s^2 = h^2 + \left(\frac{b}{2}\right)^2
\]
To solve for \(h\):
\[
h = \sqrt{s^2 - \left(\frac{b}{2}\right)^2}
\]
Once you have the values, you can plug them into their respective formulas to find the surface area and volume.
#### Input Fields:
- **Surface Area**:
- A text box where the calculated surface area will be entered (in square inches).
- **Volume**:
- A text box where the calculated volume will be entered (in cubic inches).
#### Help Options:
- **Question Help**:
- Video: A video
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