Volume Calculations: Cubes, Rectangular Prisms, Cylinders, and Spheres

Volume Calculations: Comprehensive Guide

Introduction to Volume

Volume is the amount of space occupied by a three-dimensional object. It is measured in cubic units such as cubic centimeters (cm³), cubic meters (m³), cubic inches (in³), or cubic feet (ft³).

Key Concept:

The volume of an object tells us how much space it occupies or how much substance it can contain.

1. Volume of a Cube

Formula

V = s³

Where:

V = Volume
s = Length of each side

Solution Method

Identify the length of one side of the cube.
Cube this value (multiply it by itself three times).
Add the appropriate cubic units.

Examples

Example 1: Basic Calculation

Problem: Find the volume of a cube with side length 5 cm.

Solution:

Given: s = 5 cm
Volume = s³ = (5 cm)³ = 5 cm × 5 cm × 5 cm = 125 cm³

Answer: The volume of the cube is 125 cm³.

Example 2: Finding Side Length

Problem: If a cube has a volume of 27 m³, what is the length of each side?

Solution:

Given: V = 27 m³
V = s³
27 m³ = s³
s = ∛27 m³ = 3 m

Answer: The length of each side is 3 m.

Example 3: Real-world Application

Problem: A cubical water tank has sides of length 2.5 meters. How many liters of water can it hold? (Note: 1 m³ = 1000 liters)

Solution:

Given: s = 2.5 m
Volume = s³ = (2.5 m)³ = 15.625 m³
Converting to liters: 15.625 m³ × 1000 liters/m³ = 15,625 liters

Answer: The tank can hold 15,625 liters of water.

Common Mistake:

Remember that cubing a number is different from multiplying by 3. For example, 4³ = 4 × 4 × 4 = 64, not 4 × 3 = 12.

2. Volume of a Rectangular Prism

Formula

V = l × w × h

Where:

V = Volume
l = Length
w = Width
h = Height

Solution Method

Identify the length, width, and height of the rectangular prism.
Multiply these three values together.
Add the appropriate cubic units.

Alternative Method:

You can also find the volume by calculating the area of the base (length × width) and then multiplying by the height: V = (l × w) × h

Examples

Example 1: Basic Calculation

Problem: Find the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Solution:

Given: l = 8 cm, w = 5 cm, h = 3 cm
Volume = l × w × h = 8 cm × 5 cm × 3 cm = 120 cm³

Answer: The volume of the rectangular prism is 120 cm³.

Example 2: Finding Missing Dimension

Problem: A rectangular prism has a volume of 240 in³. If the length is 10 in and the width is 6 in, what is the height?

Solution:

Given: V = 240 in³, l = 10 in, w = 6 in
V = l × w × h
240 in³ = 10 in × 6 in × h
240 in³ = 60 in² × h
h = 240 in³ ÷ 60 in² = 4 in

Answer: The height of the rectangular prism is 4 in.

Example 3: Real-world Application

Problem: A moving box has internal dimensions of 45 cm by 35 cm by 30 cm. How many cubic decimeters of items can it hold? (Note: 1 dm = 10 cm)

Solution:

Given: l = 45 cm, w = 35 cm, h = 30 cm
Volume = l × w × h = 45 cm × 35 cm × 30 cm = 47,250 cm³
Converting to dm³: 47,250 cm³ ÷ 1000 = 47.25 dm³

Answer: The box can hold 47.25 dm³ of items.

Example 4: Comparing Volumes

Problem: Box A has dimensions 12 cm × 8 cm × 6 cm. Box B has dimensions 9 cm × 9 cm × 9 cm. Which box has the greater volume and by how much?

Solution:

Volume of Box A = 12 cm × 8 cm × 6 cm = 576 cm³
Volume of Box B = 9 cm × 9 cm × 9 cm = 729 cm³
Difference = 729 cm³ - 576 cm³ = 153 cm³

Answer: Box B has the greater volume by 153 cm³.

Common Mistake:

Be careful with unit conversions when dimensions are given in different units. Always convert to the same unit before calculating volume.

3. Volume of a Cylinder

Formula

V = πr²h

Where:

V = Volume
π = Pi (approximately 3.14159)
r = Radius of the circular base
h = Height of the cylinder

Solution Method

Identify the radius (r) of the circular base and the height (h) of the cylinder.
Square the radius.
Multiply by π (use 3.14159 or the π button on your calculator).
Multiply by the height.
Add the appropriate cubic units.

Alternative Method:

You can also find the volume by calculating the area of the circular base (πr²) and then multiplying by the height: V = (πr²) × h

Examples

Example 1: Basic Calculation

Problem: Find the volume of a cylinder with radius 4 cm and height 10 cm. Use π ≈ 3.14.

Solution:

Given: r = 4 cm, h = 10 cm
Volume = πr²h = 3.14 × (4 cm)² × 10 cm
= 3.14 × 16 cm² × 10 cm
= 502.4 cm³

Answer: The volume of the cylinder is 502.4 cm³.

Example 2: Using Diameter

Problem: A cylindrical container has a diameter of 14 cm and a height of 20 cm. Find its volume using π ≈ 3.14159.

Solution:

Given: diameter = 14 cm, h = 20 cm
Radius = diameter ÷ 2 = 14 cm ÷ 2 = 7 cm
Volume = πr²h = 3.14159 × (7 cm)² × 20 cm
= 3.14159 × 49 cm² × 20 cm
= 3078.76 cm³

Answer: The volume of the cylindrical container is approximately 3078.76 cm³.

Example 3: Finding Height

Problem: A cylindrical tank has a volume of 1000 π cm³ and a radius of 5 cm. Find its height.

Solution:

Given: V = 1000π cm³, r = 5 cm
V = πr²h
1000π cm³ = π × (5 cm)² × h
1000π cm³ = π × 25 cm² × h
1000π cm³ = 25π cm² × h
h = 1000π cm³ ÷ (25π cm²) = 40 cm

Answer: The height of the cylindrical tank is 40 cm.

Example 4: Real-world Application

Problem: A water storage tank is cylindrical with a radius of 1.5 m and a height of 3 m. How many liters of water can it hold? (Note: 1 m³ = 1000 liters)

Solution:

Given: r = 1.5 m, h = 3 m
Volume = πr²h = 3.14159 × (1.5 m)² × 3 m
= 3.14159 × 2.25 m² × 3 m
= 21.21 m³
Converting to liters: 21.21 m³ × 1000 liters/m³ = 21,210 liters

Answer: The tank can hold approximately 21,210 liters of water.

Common Mistake:

Don't confuse diameter with radius. The radius is half the diameter. If you're given the diameter, divide by 2 to get the radius before using the volume formula.

4. Volume of a Sphere

Formula

V = (4/3)πr³

Where:

V = Volume
π = Pi (approximately 3.14159)
r = Radius of the sphere

Solution Method

Identify the radius (r) of the sphere.
Cube the radius (r³).
Multiply by π (use 3.14159 or the π button on your calculator).
Multiply by 4/3.
Add the appropriate cubic units.

Examples

Example 1: Basic Calculation

Problem: Find the volume of a sphere with radius 6 cm. Use π ≈ 3.14.

Solution:

Given: r = 6 cm
Volume = (4/3)πr³ = (4/3) × 3.14 × (6 cm)³
= (4/3) × 3.14 × 216 cm³
= 4.19 × 216 cm³
= 904.32 cm³

Answer: The volume of the sphere is approximately 904.32 cm³.

Example 2: Using Diameter

Problem: A spherical ball has a diameter of 10 inches. Calculate its volume using π ≈ 3.14159.

Solution:

Given: diameter = 10 inches
Radius = diameter ÷ 2 = 10 inches ÷ 2 = 5 inches
Volume = (4/3)πr³ = (4/3) × 3.14159 × (5 inches)³
= (4/3) × 3.14159 × 125 inches³
= 4.19 × 125 inches³
= 523.6 inches³

Answer: The volume of the spherical ball is approximately 523.6 cubic inches.

Example 3: Finding Radius

Problem: A sphere has a volume of 288π cm³. Find its radius.

Solution:

Given: V = 288π cm³
V = (4/3)πr³
288π cm³ = (4/3)πr³
288 = (4/3)r³
288 × (3/4) = r³
216 = r³
r = ∛216 = 6 cm

Answer: The radius of the sphere is 6 cm.

Example 4: Real-world Application

Problem: A spherical water tank has a radius of 2 meters. How many liters of water can it hold? (Note: 1 m³ = 1000 liters)

Solution:

Given: r = 2 m
Volume = (4/3)πr³ = (4/3) × 3.14159 × (2 m)³
= (4/3) × 3.14159 × 8 m³
= 33.51 m³
Converting to liters: 33.51 m³ × 1000 liters/m³ = 33,510 liters

Answer: The tank can hold approximately 33,510 liters of water.

Common Mistake:

Don't forget the fraction 4/3 in the formula. The volume of a sphere is not simply πr³, but (4/3)πr³.

Comparison of Volume Formulas

Shape	Formula	Variables	When to Use
Cube	V = s³	s = side length	When all sides are equal
Rectangular Prism	V = l × w × h	l = length w = width h = height	When the shape has three dimensions that might be different
Cylinder	V = πr²h	r = radius of circular base h = height	When the shape has circular bases
Sphere	V = (4/3)πr³	r = radius	When the shape is completely round in all directions

Key Takeaway:

Notice that each formula involves multiplying all the relevant dimensions. For shapes with circular components, π is included in the formula.

Volume Unit Conversions

Metric System

1 cubic meter (m³) = 1,000 cubic decimeters (dm³) = 1,000,000 cubic centimeters (cm³)
1 cubic decimeter (dm³) = 1,000 cubic centimeters (cm³)
1 cubic centimeter (cm³) = 1,000 cubic millimeters (mm³)
1 liter (L) = 1 cubic decimeter (dm³) = 1,000 cubic centimeters (cm³) = 1,000 milliliters (mL)

Imperial/US System

1 cubic yard (yd³) = 27 cubic feet (ft³)
1 cubic foot (ft³) = 1,728 cubic inches (in³)
1 gallon (gal) = 231 cubic inches (in³) = 4 quarts (qt)
1 quart (qt) = 2 pints (pt)
1 pint (pt) = 16 fluid ounces (fl oz)

Cross-System Conversions

1 cubic meter (m³) ≈ 35.31 cubic feet (ft³) ≈ 1.308 cubic yards (yd³)
1 liter (L) ≈ 0.264 US gallons (gal) ≈ 1.057 US quarts (qt)
1 cubic inch (in³) ≈ 16.39 cubic centimeters (cm³)
1 US gallon (gal) ≈ 3.785 liters (L)

Key Conversion Tips:

When converting between cubic units in the metric system, remember that each step up or down the scale involves a factor of 1000, not 10, because you're working in three dimensions.

Additional Practice Problems

Problem 1: Cube

A cube has a side length of 9 meters. Find its volume.

Problem 2: Rectangular Prism

A rectangular fish tank has a length of 60 cm, width of 30 cm, and height of 40 cm. How many liters of water can it hold? (1000 cm³ = 1 liter)

Problem 3: Cylinder

A cylindrical water tower has a diameter of 20 meters and a height of 35 meters. What is its capacity in cubic meters? Use π ≈ 3.14.

Problem 4: Sphere

The Earth has an approximate radius of 6,371 kilometers. Calculate its volume in cubic kilometers. Use π ≈ 3.14159.

Problem 5: Combined Shapes

A solid consists of a cube with side length 8 cm with a cylindrical hole drilled through the center. The cylindrical hole has a radius of 2 cm and goes through the entire cube. Find the volume of the resulting solid. Use π ≈ 3.14.