Three-Dimensional (Volume) Unit Conversion

Volume is a three-dimensional space that can be measured in cubic units.

We can relate volume units with length units because volume is derived from length measurements. For example, a cube that is 10 meters on each side has a volume of 1,000 cubic meters (m³):

Metric Prefixes for Volume Units

Therefore, volume prefixes work the same way as length prefixes but it goes in thousands instead of tens.

Volume UnitVolume FactorExample Conversion
Mm³(10³)⁶ = 10¹⁸1 Mm³ = 10¹⁸ m³
km³(10³)³ = 10⁹1 km³ = 10⁹ m³
hm³(10³)² = 10⁶1 hm³ = 10⁶ m³
dam³(10³)¹ = 10³1 dam³ = 10³ m³
m³(10³)⁰ = 10⁰Base unit
dm³(10³)⁻¹ = 10⁻³1 dm³ = 10⁻³ m³
cm³(10³)⁻² = 10⁻⁶1 cm³ = 10⁻⁶ m³
mm³(10³)⁻³ = 10⁻⁹1 mm³ = 10⁻⁹ m³
μm³(10³)⁻⁶ = 10⁻¹⁸1 μm³ = 10⁻¹⁸ m³
Desired ConversionEquivalence RelationshipFormula and Calculation
2.5 km³ to m³1 km³ = 10⁹ m³$2.5 \text{ km}^3 \times \frac{10^9 \text{ m}^3}{1 \text{ km}^3} = 2.5 \times 10^9 \text{ m}^3$
0.8 hm³ to m³1 hm³ = 10⁶ m³$0.8 \text{ hm}^3 \times \frac{10^6 \text{ m}^3}{1 \text{ hm}^3} = 8 \times 10^5 \text{ m}^3$
150 dam³ to m³1 dam³ = 10³ m³$150 \text{ dam}^3 \times \frac{10^3 \text{ m}^3}{1 \text{ dam}^3} = 1.5 \times 10^5 \text{ m}^3$
3.2 m³ to dm³1 m³ = 10³ dm³$3.2 \text{ m}^3 \times \frac{10^3 \text{ dm}^3}{1 \text{ m}^3} = 3.2 \times 10^3 \text{ dm}^3$
0.75 m³ to cm³1 m³ = 10⁶ cm³$0.75 \text{ m}^3 \times \frac{10^6 \text{ cm}^3}{1 \text{ m}^3} = 7.5 \times 10^5 \text{ cm}^3$
4.6 m³ to mm³1 m³ = 10⁹ mm³$4.6 \text{ m}^3 \times \frac{10^9 \text{ mm}^3}{1 \text{ m}^3} = 4.6 \times 10^9 \text{ mm}^3$
2.8 km³ to cm³1 km³ = 10⁹ m³, 1 m³ = 10⁶ cm³$2.8 \text{ km}^3 \times \frac{10^9 \text{ m}^3}{1 \text{ km}^3} \times \frac{10^6 \text{ cm}^3}{1 \text{ m}^3} = 2.8 \times 10^{15} \text{ cm}^3$
0.35 hm³ to mm³1 hm³ = 10⁶ m³, 1 m³ = 10⁹ mm³$0.35 \text{ hm}^3 \times \frac{10^6 \text{ m}^3}{1 \text{ hm}^3} \times \frac{10^9 \text{ mm}^3}{1 \text{ m}^3} = 3.5 \times 10^{14} \text{ mm}^3$
120 dam³ to dm³1 dam³ = 10³ m³, 1 m³ = 10³ dm³$120 \text{ dam}^3 \times \frac{10^3 \text{ m}^3}{1 \text{ dam}^3} \times \frac{10^3 \text{ dm}^3}{1 \text{ m}^3} = 1.2 \times 10^8 \text{ dm}^3$

Solved Examples

Example 1: Convert 0.5 km³ to cm³

Method 1: Direct conversion

1 km = 1,000 m = 100,000 cm

So 1 km³ = (100,000 cm)³ = 10¹⁵ cm³

$0.5 \text{ km}^3 \times \frac{10^{15} \text{ cm}^3}{1 \text{ km}^3} = 5 \times 10^{14} \text{ cm}^3$

Method 2: Step-by-step

$0.5 \text{ km}^3 \times \frac{10^9 \text{ m}^3}{1 \text{ km}^3} \times \frac{10^6 \text{ cm}^3}{1 \text{ m}^3} = 0.5 \times 10^{15} \text{ cm}^3 = 5 \times 10^{14} \text{ cm}^3$

Result: 0.5 km³ = 5 × 10¹⁴ cm³ = 500,000,000,000,000 cm³

Example 2: A rectangular water tank is 25 m long and 12.5 m wide and 2 m high. Calculate its volume in liters and cubic feet.

Step 1: Calculate volume in m³

$ \text{Volume} = 25 \text{ m} \times 12.5 \text{ m} \times 2 \text{ m} = 625 \text{ m}^3$

Step 2: Convert to liters

$625 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 625,000 \text{ L}$

Step 3: Convert to cubic feet

1 m³ ≈ 35.31 ft³

$625 \text{ m}^3 \times \frac{35.31 \text{ ft}^3}{1 \text{ m}^3} ≈ 22,069 \text{ ft}^3$

Result: Tank volume = 625 m³ = 625,000 L ≈ 22,069 ft³

Example 3: A swimming pool is 15 m long and 800 cm wide and 150 cm deep. What is its volume in liters?

Method 1: Convert everything to m first

Length: 15 m

Width: 800 cm = 8 m

Depth: 150 cm = 1.5 m

$ \text{Volume} = 15 \text{ m} \times 8 \text{ m} \times 1.5 \text{ m} = 180 \text{ m}^3$

Method 2: Convert to liters

$180 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 180,000 \text{ L}$

Result: Pool volume = 180,000 L

Example 4: A cylindrical tank has a diameter of 200 cm and height 150 cm. Find its volume in liters and gallons.

Step 1: Find radius and calculate volume

Radius = 200 cm ÷ 2 = 100 cm = 1 m, Height = 150 cm = 1.5 m

$V = \pi r^2 h = 3.14159 \times (1 \text{ m})^2 \times 1.5 \text{ m} = 4.71 \text{ m}^3$

Step 2: Convert to liters

$4.71 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 4,710 \text{ L}$

Step 3: Convert to gallons

$4,710 \text{ L} \times \frac{1 \text{ gal}}{3.785 \text{ L}} ≈ 1,244 \text{ gal}$

Result: Tank volume = 4.71 m³ = 4,710 L ≈ 1,244 gal

Example 5: A cube has sides of 20 cm. What is its volume in liters and milliliters?

Step 1: Calculate volume in cm³

$V = (20 \text{ cm})^3 = 8,000 \text{ cm}^3$

Step 2: Convert to liters

Since 1 L = 1,000 cm³:

$8,000 \text{ cm}^3 \times \frac{1 \text{ L}}{1,000 \text{ cm}^3} = 8 \text{ L}$

Step 3: Convert to milliliters

Since 1 cm³ = 1 mL:

$8,000 \text{ cm}^3 = 8,000 \text{ mL}$

Result: Cube volume = 8,000 cm³ = 8 L = 8,000 mL

1. Convert 0.8 m³ to liters: (1 point)

2. Convert 3.2 dm³ to cm³: (1 point)

3. Convert 2 500 mL to liters: (1 point)

4. Convert 5 cm³ to mm³: (1 point)

5. Volume conversions with metric prefixes: (20 points)

a ) Convert 2.5 m³ to L:

b ) Convert 750 mL to L:

c ) Convert 1.2 dm³ to cm³:

d ) Convert 3 500 cm³ to L:

e ) Convert 0.8 L to mL:

f ) Convert 4.5 × 10⁶ mm³ to dm³:

g ) Convert 0.25 m³ to cm³:

h ) Convert 15 L to dm³:

6. Imperial and mixed volume conversions: (12 points)

a ) Convert 2 gallons to liters (1 gallon = 3.785 L):

b ) Convert 1 cubic foot to liters (1 ft = 30.48 cm):

c ) Convert 3 quarts to mL (1 quart = 0.946 L):

d ) Convert 50 L to cubic feet (1 ft³ ≈ 28.3 L):

7. Geometric volume calculations: (15 points)

a ) A cube has sides of 20 cm. What is its volume in liters?

b ) A cylinder has radius 5 cm and height 10 cm. What is its volume in mL (use π ≈ 3.14)?

c ) A rectangular tank is 2 m × 1.5 m × 80 cm. What is its volume in liters?

d ) A sphere has radius 6 cm. What is its volume in cm³ (use π ≈ 3.14, V = 4/3πr³)?

e ) A triangular prism has base area 15 cm² and height 8 cm. What is its volume in mL?