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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 Unit | Volume Factor | Example Conversion |
|---|---|---|
| Mm³ | (10³)⁶ = 10¹⁸ | 1 Mm³ = 10¹⁸ m³ |
| km³ | (10³)³ = 10⁹ | 1 km³ = 10⁹ m³ |
| hm³ | (10³)² = 10⁶ | 1 hm³ = 10⁶ m³ |
| dam³ | (10³)¹ = 10³ | 1 dam³ = 10³ m³ |
| m³ | (10³)⁰ = 10⁰ | Base unit |
| dm³ | (10³)⁻¹ = 10⁻³ | 1 dm³ = 10⁻³ m³ |
| cm³ | (10³)⁻² = 10⁻⁶ | 1 cm³ = 10⁻⁶ m³ |
| mm³ | (10³)⁻³ = 10⁻⁹ | 1 mm³ = 10⁻⁹ m³ |
| μm³ | (10³)⁻⁶ = 10⁻¹⁸ | 1 μm³ = 10⁻¹⁸ m³ |
| Desired Conversion | Equivalence Relationship | Formula and Calculation |
|---|---|---|
| 2.5 km³ to m³ | 1 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 m³ | 1 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 m³ | 1 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 dm³ | 1 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 cm³ | 1 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 mm³ | 1 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 cm³ | 1 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 mm³ | 1 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 dm³ | 1 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