What is the volume and surface area of a hollow cylinder outer diameter 60 mm, inner diameter 40 mm and length 200 mm?

To calculate the **volume** and **surface area** of a **hollow cylinder**, we'll need to use the following formulas:

### 1. **Volume of a Hollow Cylinder:**
The volume of a hollow cylinder is given by the difference in the volumes of two solid cylinders (outer and inner cylinders):

\[
V = \pi h \left( R_{\text{outer}}^2 - R_{\text{inner}}^2 \right)
\]

Where:
- \( h \) = height (or length) of the cylinder = 200 mm,
- \( R_{\text{outer}} \) = outer radius = \( \frac{60}{2} = 30 \, \text{mm} \),
- \( R_{\text{inner}} \) = inner radius = \( \frac{40}{2} = 20 \, \text{mm} \).

Now, substituting the values:

\[
V = \pi (200) \left( 30^2 - 20^2 \right) = \pi (200) \left( 900 - 400 \right) = \pi (200) \times 500
\]

\[
V = 100,000 \pi \, \text{mm}^3 \approx 314,159.27 \, \text{mm}^3
\]

### 2. **Surface Area of a Hollow Cylinder:**
The total surface area of a hollow cylinder includes:
- The lateral surface areas of both the outer and inner cylinders,
- The surface areas of the two circular faces (rings).

The formula for the surface area is:

\[
A = 2 \pi h (R_{\text{outer}} + R_{\text{inner}}) + 2 \pi \left( R_{\text{outer}}^2 - R_{\text{inner}}^2 \right)
\]

Now, substituting the values:

\[
A = 2 \pi (200) (30 + 20) + 2 \pi (900 - 400)
\]

\[
A = 2 \pi (200) \times 50 + 2 \pi \times 500 = 20,000 \pi + 1,000 \pi = 21,000 \pi
\]

\[
A \approx 65,973.45 \, \text{mm}^2
\]

### Final Answers:
- **Volume**: \( \approx 314,159.27 \, \text{mm}^3 \)
- **Surface Area**: \( \approx 65,973.45 \, \text{mm}^2 \)

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the volume change if the length of the cylinder doubled?
2. What if the outer diameter was reduced by 10 mm, how would that affect the volume?
3. Can you calculate the lateral surface area separately for the inner and outer parts?
4. How does the surface area compare to that of a solid cylinder with the same outer dimensions?
5. How would you calculate the mass of the hollow cylinder if the material density is known?

**Tip:** To convert between cubic millimeters and liters, remember that \( 1 \, \text{liter} = 1,000,000 \, \text{mm}^3 \).

Calculate the volume and surface area of a hollow cylinder with an outer diameter of 60 mm, inner diameter of 40 mm, and length of 200 mm. This geometry problem covers the key formulas and provides step-by-step solutions for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation