Volume Cylinder Worksheet Answer Key

Mastering Cylinder Volume: A Comprehensive Worksheet and Answer Key

Understanding how to calculate the volume of a cylinder is a fundamental skill in geometry, crucial for various applications in engineering, architecture, and everyday life. This comprehensive guide provides a detailed explanation of the formula, walks you through step-by-step examples with solutions, and offers a worksheet with diverse problems to test your understanding. We'll cover everything from basic calculations to more complex scenarios involving combined shapes and real-world applications. By the end, you'll be confident in tackling any cylinder volume problem that comes your way.

Understanding Cylinder Volume: The Basics

A cylinder is a three-dimensional shape with two parallel circular bases connected by a curved surface. The volume of a cylinder represents the amount of space it occupies. Calculating this volume is straightforward, relying on a simple yet powerful formula:

Volume (V) = πr²h

Where:

• V represents the volume of the cylinder.
• π (pi) is a mathematical constant, approximately equal to 3.14159. For most calculations, using 3.14 is sufficient.
• r represents the radius of the circular base (half the diameter).
• h represents the height of the cylinder.

This formula essentially tells us to find the area of the circular base (πr²) and then multiply it by the height (h) to determine the total volume.

Step-by-Step Calculation Process

Let's break down the calculation process into easy-to-follow steps:

1. Identify the radius (r) and height (h): Carefully examine the given information in the problem. Ensure you understand whether you're given the radius or diameter. Remember, the radius is half the diameter.

2. Square the radius (r²): Multiply the radius by itself (r x r).

3. Multiply by π: Multiply the squared radius by π (approximately 3.14).

4. Multiply by the height (h): Finally, multiply the result from step 3 by the height of the cylinder.

5. State your answer: Always include the appropriate units (cubic centimeters, cubic meters, cubic inches, etc.) in your final answer.

Examples and Worked Solutions

Let's solidify our understanding with some examples.

Example 1: Basic Calculation

A cylinder has a radius of 5 cm and a height of 10 cm. Calculate its volume.

Solution:

1. r = 5 cm, h = 10 cm

2. r² = 5 cm × 5 cm = 25 cm²

3. πr² = 3.14 × 25 cm² = 78.5 cm²

4. V = πr²h = 78.5 cm² × 10 cm = 785 cm³

Therefore, the volume of the cylinder is 785 cubic centimeters.

Example 2: Using Diameter

A cylinder has a diameter of 12 inches and a height of 8 inches. Find its volume.

Solution:

1. Diameter = 12 inches, so r = 12 inches / 2 = 6 inches. h = 8 inches

2. r² = 6 inches × 6 inches = 36 inches²

3. πr² = 3.14 × 36 inches² = 113.04 inches²

4. V = πr²h = 113.04 inches² × 8 inches = 904.32 inches³

The volume of the cylinder is approximately 904.32 cubic inches.

Example 3: Problem Solving with Multiple Steps

A cylindrical water tank has a volume of 1570 cubic meters. If its height is 10 meters, what is the radius of the tank?

Solution:

This problem requires us to rearrange the volume formula to solve for the radius:

1. V = 1570 m³, h = 10 m

2. V = πr²h => r² = V / (πh)

3. r² = 1570 m³ / (3.14 × 10 m) = 50 m²

4. r = √50 m² ≈ 7.07 m

The radius of the water tank is approximately 7.07 meters.

Cylinder Volume Worksheet

Now, let's put your knowledge to the test! Solve the following problems, showing your work for each. Remember to include units in your answers.

Problem 1: A cylinder has a radius of 3 cm and a height of 7 cm. Calculate its volume.

Problem 2: A cylindrical can has a diameter of 10 inches and a height of 12 inches. What is its volume?

Problem 3: A water pipe has a radius of 2.5 cm and a length of 5 meters. Calculate the volume of water it can hold. (Remember to convert units to be consistent).

Problem 4: A cylindrical silo has a volume of 5000 cubic feet. If its radius is 10 feet, what is its height?

Problem 5: Two cylinders have the same height of 15cm. Cylinder A has a radius of 4cm, and Cylinder B has a radius of 6cm. What is the difference in their volumes?

Problem 6: A cylindrical container has a volume of 2000 cubic centimeters. If its height is doubled, what will be its new volume?

Problem 7: A cylindrical container with a radius of 5 cm is filled with water to a height of 10 cm. If more water is added, increasing the water level to 15 cm, how much additional water was added?

Cylinder Volume Worksheet Answer Key

Problem 1: V = πr²h = 3.14 × (3cm)² × 7cm = 197.82 cm³

Problem 2: r = 10 inches / 2 = 5 inches; V = πr²h = 3.14 × (5 inches)² × 12 inches = 942 inches³

Problem 3: Convert 5 meters to centimeters (500 cm); V = πr²h = 3.14 × (2.5 cm)² × 500 cm = 9812.5 cm³

Problem 4: V = πr²h => h = V / (πr²) = 5000 ft³ / (3.14 × (10 ft)²) ≈ 15.92 ft

Problem 5: Volume A = π(4cm)²(15cm) ≈ 753.6 cm³; Volume B = π(6cm)²(15cm) ≈ 1695.6 cm³; Difference = 1695.6 cm³ - 753.6 cm³ = 942 cm³

Problem 6: Let the original volume be V₁ = πr²h. Doubling the height gives V₂ = πr²(2h) = 2πr²h = 2V₁. The new volume will be 4000 cubic centimeters.

Problem 7: Initial volume V₁ = π(5cm)²(10cm) ≈ 785 cm³; Final volume V₂ = π(5cm)²(15cm) ≈ 1177.5 cm³; Additional water = V₂ - V₁ ≈ 392.5 cm³

Frequently Asked Questions (FAQ)

Q: What if I'm given the circumference instead of the radius or diameter?

A: Remember that the circumference (C) of a circle is given by C = 2πr. If you're given the circumference, you can solve for the radius (r = C / 2π) and then use the standard volume formula.

Q: How do I calculate the volume of a cylinder with an oblique (slanted) height?

A: The formula V = πr²h only applies to right cylinders (where the height is perpendicular to the base). For oblique cylinders, you need to use the perpendicular height, not the slanted height. You may need additional information or trigonometry to find the perpendicular height.

Q: Can I use this formula for other shapes?

A: No, this formula is specifically for cylinders. Different shapes (cones, spheres, prisms) have their own unique volume formulas.

Q: What are some real-world applications of cylinder volume calculations?

A: Cylinder volume calculations are used extensively in various fields. Examples include determining the capacity of water tanks, calculating the amount of material needed for pipes, designing engine cylinders, and even estimating the volume of tree trunks.

Conclusion

Mastering the calculation of cylinder volume is a valuable skill with numerous practical applications. By understanding the formula, following the step-by-step process, and practicing with various problems, you can confidently tackle any cylinder volume challenge. Remember to always double-check your units and use the correct formula for the specific type of cylinder you are working with. Consistent practice is key to building a strong foundation in geometry and problem-solving. So, keep practicing, and you'll become a cylinder volume expert in no time!