Determine the volume and the surface area of the half-torus shown in the figure. R
Determine the volume and the surface area of the half-torus shown in the figure. R
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![**Title: Calculating Volume and Surface Area of a Half-Torus**
**Objective:**
Learn how to determine the volume and surface area of a half-torus using the formulas provided.
**Introduction:**
A torus is a three-dimensional shape that resembles a doughnut. It has two radii:
1. \( R \) - the distance from the center of the tube to the center of the torus.
2. \( r \) - the radius of the tube itself.
In this exercise, we will focus on a half-torus.
**Visual Representation:**
The image shows a half-torus intersecting the x-y plane. The major radius \( R \) is shown as the distance from the origin to the center of the torus, while the minor radius \( r \) is the radius of the circular cross-section of the torus.
**Formulas:**
- **Volume of the Half-Torus:**
\[
V = \pi^2 R r^2
\]
To calculate the volume, plug the known values of \( R \) and \( r \) into the formula and solve.
- **Surface Area of the Half-Torus:**
\[
A = \pi^2 R r
\]
Use the above formula for surface area to find the half-torus's surface area.
**Application:**
Fill in the blanks with the computed values for volume and surface area based on the given radii.
This exercise aims to solidify your understanding of geometric volume and surface area calculations for complex shapes like the torus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffc9e826-c865-4bea-8e1f-6dcccd31bfb0%2F88f1fec1-f02c-47c7-8255-53265672f5c6%2F8tu212k_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Volume and Surface Area of a Half-Torus**
**Objective:**
Learn how to determine the volume and surface area of a half-torus using the formulas provided.
**Introduction:**
A torus is a three-dimensional shape that resembles a doughnut. It has two radii:
1. \( R \) - the distance from the center of the tube to the center of the torus.
2. \( r \) - the radius of the tube itself.
In this exercise, we will focus on a half-torus.
**Visual Representation:**
The image shows a half-torus intersecting the x-y plane. The major radius \( R \) is shown as the distance from the origin to the center of the torus, while the minor radius \( r \) is the radius of the circular cross-section of the torus.
**Formulas:**
- **Volume of the Half-Torus:**
\[
V = \pi^2 R r^2
\]
To calculate the volume, plug the known values of \( R \) and \( r \) into the formula and solve.
- **Surface Area of the Half-Torus:**
\[
A = \pi^2 R r
\]
Use the above formula for surface area to find the half-torus's surface area.
**Application:**
Fill in the blanks with the computed values for volume and surface area based on the given radii.
This exercise aims to solidify your understanding of geometric volume and surface area calculations for complex shapes like the torus.
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