Solved Exercises on Volume of Cylinders in Grade 8

Master volume of cylinders: radius, height, diameter calculations, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Cylinder Volume
Exercise 1
Find the volume of a cylinder with radius 5 cm and height 12 cm. Use π ≈ 3.14.
Definition:

Volume of Cylinder: The amount of space inside a cylindrical container. The formula is V = πr²h, where r is the radius of the circular base and h is the height.

Volume Calculation Method:
  1. Identify the radius (r) and height (h)
  2. Apply the formula V = πr²h
  3. Substitute the values and calculate
  4. Include the correct units (cubic units)
Given Values
r = 5 cm, h = 12 cm
Formula
V = πr²h
Result
V = 942 cm³
Step 1: Write down the formula

V = πr²h

Step 2: Substitute the known values

V = π × (5)² × 12

V = π × 25 × 12

Step 3: Calculate

V = 3.14 × 25 × 12

V = 3.14 × 300

V = 942 cm³

V = 942 cm³
Final answer:

The volume of the cylinder is 942 cubic centimeters.

Applied rules:

• Volume Formula: V = πr²h for cylinders

• Units: Volume is measured in cubic units

• Calculation Order: Exponents first, then multiplication

2 Diameter to Radius Conversion
Exercise 2
A cylindrical water tank has a diameter of 8 feet and a height of 10 feet. Find its volume. Use π ≈ 3.14.
Definition:

Diameter and Radius Relationship: The diameter (d) is twice the radius (r), so d = 2r or r = d/2. Always convert diameter to radius before using the volume formula.

Given Values
d = 8 ft, h = 10 ft
Convert to Radius
r = 4 ft
Volume
V = 502.4 ft³
Step 1: Convert diameter to radius

r = d/2 = 8/2 = 4 feet

Step 2: Apply the volume formula

V = πr²h = π × (4)² × 10

Step 3: Calculate

V = 3.14 × 16 × 10

V = 3.14 × 160

V = 502.4 ft³

V = 502.4 ft³
Final answer:

The volume of the water tank is 502.4 cubic feet.

Applied rules:

• Diameter to Radius: r = d/2

• Volume Formula: V = πr²h

• Unit Consistency: Keep all measurements in the same units

3 Finding Missing Dimensions
Exercise 3
The volume of a cylinder is 1570 cubic inches and its height is 10 inches. Find the radius of the cylinder. Use π ≈ 3.14.
Definition:

Algebraic Manipulation: When solving for missing dimensions, rearrange the volume formula algebraically. From V = πr²h, we get r² = V/(πh) and r = √[V/(πh)].

Given Values
V = 1570 in³, h = 10 in
Formula Rearranged
r² = V/(πh)
Radius
r = 7.07 in
Step 1: Start with the volume formula

V = πr²h

Step 2: Solve for r²

1570 = π × r² × 10

1570 = 3.14 × r² × 10

1570 = 31.4 × r²

r² = 1570/31.4

r² = 50

Step 3: Solve for r

r = √50 ≈ 7.07 inches

r ≈ 7.07 inches
Final answer:

The radius of the cylinder is approximately 7.07 inches.

Applied rules:

• Algebraic Rearrangement: Isolate the unknown variable

• Square Root: Take the positive root since radius is positive

• Verification: Check by substituting back into original formula

Rules and methods, laws,...
\(V = \pi r^2 h\)
Volume of Cylinder
Volume Formula
V = πr²h
Base area × height
Diameter to Radius
r = d/2
Always convert first
Base Area
A = πr²
Area of circular base
Finding Radius
r = √(V/πh)
When volume and height known
Finding Height
h = V/πr²
When volume and radius known
Base Area: The area of the circular base is πr²
Volume Concept: Volume = Base area × height for any prism or cylinder
Key definitions:

Cylinder: A three-dimensional shape with two parallel circular bases connected by a curved surface.

Radius (r): The distance from the center of the circular base to its edge.

Diameter (d): The distance across the circular base, passing through the center. d = 2r.

Height (h): The perpendicular distance between the two circular bases.

Volume: The amount of space inside a three-dimensional object, measured in cubic units.

Complete methodology:
  1. Identify given information: Determine which measurements are provided
  2. Convert units if needed: Ensure all measurements are in the same units
  3. Apply the correct formula: Use V = πr²h
  4. Perform calculations: Follow order of operations
  5. Include units: Always express the answer in cubic units
Tip 1: Always convert diameter to radius before using the formula.
Tip 2: Remember that volume is always expressed in cubic units.
Tip 3: Double-check your calculations, especially with π approximations.
Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A soup can has a diameter of 7 cm and a height of 10 cm. If 1 cm³ = 1 mL, how many liters of soup can the can hold? Use π ≈ 3.14.
Definition:

Real-World Conversion: 1 cm³ = 1 mL and 1000 mL = 1 L. Understanding unit conversions is crucial for practical applications.

Given Values
d = 7 cm, h = 10 cm
Convert to Radius
r = 3.5 cm
Volume
V = 384.65 mL = 0.385 L
Step 1: Find the radius

r = d/2 = 7/2 = 3.5 cm

Step 2: Calculate the volume

V = πr²h = 3.14 × (3.5)² × 10

V = 3.14 × 12.25 × 10

V = 3.14 × 122.5

V = 384.65 cm³

Step 3: Convert to liters

Since 1 cm³ = 1 mL, V = 384.65 mL

Since 1000 mL = 1 L, V = 384.65/1000 = 0.385 L

0.385 liters
Final answer:

The soup can can hold approximately 0.385 liters of soup.

Applied rules:

• Unit Conversion: 1 cm³ = 1 mL, 1000 mL = 1 L

• Diameter to Radius: r = d/2

• Volume Formula: V = πr²h

5 Comparative Analysis
Exercise 5
Cylinder A has radius 4 cm and height 6 cm. Cylinder B has radius 6 cm and height 4 cm. Which cylinder has the greater volume, and by how much? Use π ≈ 3.14.
Definition:

Comparative Analysis: Calculating volumes of different shapes to compare their capacities. Note that changing dimensions affects volume differently.

Cylinder A
V_A = 301.44 cm³
Cylinder B
V_B = 452.16 cm³
Difference
ΔV = 150.72 cm³
Step 1: Calculate volume of Cylinder A

V_A = πr²h = 3.14 × (4)² × 6

V_A = 3.14 × 16 × 6 = 301.44 cm³

Step 2: Calculate volume of Cylinder B

V_B = πr²h = 3.14 × (6)² × 4

V_B = 3.14 × 36 × 4 = 452.16 cm³

Step 3: Compare volumes

V_B > V_A

Difference = V_B - V_A = 452.16 - 301.44 = 150.72 cm³

Step 4: Analyze the result

Even though A and B have swapped dimensions (4×6 vs 6×4), Cylinder B has the greater volume because the radius is squared in the formula, making it more sensitive to changes in radius than height.

Cylinder B has greater volume by 150.72 cm³
Final answer:

Cylinder B has the greater volume by 150.72 cubic centimeters.

Applied rules:

• Volume Formula: V = πr²h for both cylinders

• Comparative Analysis: Calculate each volume separately

• Radius Effect: Since r is squared, it has a greater impact on volume than height

Volume of Cylinders Laws, Methods, and Properties
\(V = \pi r^2 h\)
Volume of Cylinder
Key definitions:

Volume of Cylinder: The measure of space occupied by a cylinder, calculated as the area of the circular base multiplied by the height. Formula: V = πr²h.

Circular Base: The flat, round surface at the top and bottom of the cylinder with area A = πr².

Curved Surface: The lateral surface connecting the two circular bases, contributing to the shape but not directly to the volume calculation.

Complete methodology:
  1. Identify measurements: Determine radius (not diameter) and height
  2. Check units: Ensure consistent units for radius and height
  3. Apply formula: V = πr²h
  4. Calculate: Square the radius first, then multiply by π and height
  5. Express answer: Include correct cubic units
Tip 1: The radius has a greater effect on volume than height since it's squared.
Tip 2: Always convert diameter to radius before using the formula.
Tip 3: For missing dimensions, rearrange the formula algebraically.
Tip 4: Remember that volume is always expressed in cubic units.
Common errors: Forgetting to convert diameter to radius, using incorrect units, mixing up radius and height, miscalculating with π, forgetting to cube the units.
Exam preparation: Practice with various units, master dimensional analysis, understand the impact of changing dimensions, solve for missing values.
Volume formulas and properties:

• Basic Formula: V = πr²h

• Base Area: A = πr²

• Diameter Conversion: r = d/2

• Finding Radius: r = √(V/πh)

• Finding Height: h = V/πr²

• Unit Relationship: 1 cm³ = 1 mL, 1000 cm³ = 1 L

• Proportional Relationships: Volume varies as the square of the radius and directly with height.

Exercise with Visualization: Volume Relationships
Exercise 6: Volume vs Radius/Height
Consider cylinders with varying dimensions:
Fixed height (h=10), varying radius: r=1, 2, 3, 4, 5
Fixed radius (r=3), varying height: h=2, 4, 6, 8, 10

Analysis: The chart shows how volume changes with radius and height.

  • Volume increases quadratically with radius (V ∝ r²)
  • Volume increases linearly with height (V ∝ h)
  • Radius has a greater impact on volume than height

Questions & Answers

Question: Why is the radius squared in the volume formula? Shouldn't it be linear like the height?

Answer: The radius is squared because the base of the cylinder is a circle, and the area of a circle is A = πr². Since volume is base area times height, we get V = πr²h.

Think of it this way:

  • The base area depends on the radius squared (A = πr²)
  • We then stack this base area up to the height h
  • So the radius affects the volume through the base area, which is why it's squared

This is why increasing the radius has a much bigger impact on volume than increasing the height. If you double the radius, volume increases by a factor of 4, but if you double the height, volume only doubles.

Question: How do I handle problems where I'm given the circumference instead of the radius?

Answer: If you're given the circumference (C), you can find the radius using the circumference formula:

C = 2πr

Solving for r: r = C/(2π)

Example: If a cylinder has circumference 12.56 cm and height 8 cm:

  • Find radius: r = 12.56/(2×3.14) = 12.56/6.28 = 2 cm
  • Then use volume formula: V = πr²h = 3.14 × 4 × 8 = 100.48 cm³

Always convert any given measurement to radius before using the volume formula!

Question: What's the difference between the surface area and volume of a cylinder? They seem related.

Answer: While both involve the cylinder's dimensions, they measure different things:

  • Volume: Measures the space inside the cylinder (what it can hold). Formula: V = πr²h. Measured in cubic units.
  • Surface Area: Measures the total area of all surfaces (the "skin" of the cylinder). Formula: SA = 2πrh + 2πr². Measured in square units.

Think of it this way:

  • Volume = how much liquid fills the cylinder
  • Surface Area = how much wrapping paper covers the cylinder

They both use π, r, and h, but in different ways and for different purposes!

Question: How do I know which value to assign to radius and height when solving problems?

Answer: The key is to identify the dimensions correctly:

  • Radius: The distance from the center of the circular base to its edge. It's half the diameter.
  • Height: The perpendicular distance between the two circular bases. It's the "length" of the cylinder.

Tips for identification:

  • Look for the circular dimension (distance across the base) - that's diameter, so divide by 2 to get radius
  • Look for the straight-line distance along the side - that's the height
  • Radius is usually smaller than height in typical cylinders

Example: A soup can with "diameter 7 cm, height 10 cm" - radius is 3.5 cm, height is 10 cm.

The radius always goes into the formula as r, and height as h!

Question: What's the difference between lateral surface area and total surface area of a cylinder?

Answer: The surface area components are:

  • Lateral Surface Area: Just the curved side of the cylinder. Formula: LSA = 2πrh
  • Total Surface Area: Lateral surface area plus both circular bases. Formula: TSA = 2πrh + 2πr²

Think of it this way:

  • Lateral = just the "wall" of the cylinder
  • Total = the wall plus the top and bottom circles

Example: For a cylinder with r=3 cm and h=5 cm:

  • Lateral SA = 2π(3)(5) = 30π cm²
  • Total SA = 30π + 2π(9) = 30π + 18π = 48π cm²

Be careful to read problems to see which surface area they're asking for!