Loading history...
Two-Dimensional (Area) Unit Conversion
An area or surface is a two-dimensional space that can be measured in square units.
We can relate surface units with length units because area is derived from length measurements. For example, a square that is 10 meters on each side has an area of 100 square meters (m²):
Metric Prefixes for Area Units
Therefore, area prefixes work the same way as length prefixes but it goes in hundreds instead of tens.
| Area Unit | Area Factor | Example Conversion |
|---|---|---|
| Mm² | (10²)⁶ = 100¹² | 1 Mm² = 10²⁴ m² |
| km² | (10²)³ = 100³ | 1 km² = 10⁶ m² |
| hm² | (10²)² = 100² | 1 hm² = 10⁴ m² |
| dam² | (10²)¹ = 100¹ | 1 dam² = 10² m² |
| m² | (10²)⁰ = 100⁰ | Base unit |
| dm² | (10²)⁻¹ = 100⁻¹ | 1 m² = 10⁻² dm² |
| cm² | (10²)⁻² = 10⁻⁴ | 1 cm² = 10⁻⁴ m² |
| mm² | (10²)⁻³ = 100⁻³ | 1 mm² = 10⁻⁶ m² |
| μm² | (10²)⁻⁶ = 100⁻¹² | 1 μm² = 10⁻²⁴ m² |
| Desired Conversion | Equivalence Relationship | Formula and Calculation |
|---|---|---|
| 2.5 km² to m² | 1 km² = 10⁶ m² | \(2.5 \text{ km}^2 \times \frac{10^6 \text{ m}^2}{1 \text{ km}^2} = 2.5 \times 10^6 \text{ m}^2\) |
| 0.8 hm² to m² | 1 hm² = 10⁴ m² | \(0.8 \text{ hm}^2 \times \frac{10^4 \text{ m}^2}{1 \text{ hm}^2} = 8 \times 10^3 \text{ m}^2\) |
| 150 dam² to m² | 1 dam² = 10² m² | \(150 \text{ dam}^2 \times \frac{10^2 \text{ m}^2}{1 \text{ dam}^2} = 1.5 \times 10^4 \text{ m}^2\) |
| 3.2 m² to dm² | 1 m² = 10² dm² | \(3.2 \text{ m}^2 \times \frac{10^2 \text{ dm}^2}{1 \text{ m}^2} = 3.2 \times 10^2 \text{ dm}^2\) |
| 0.75 m² to cm² | 1 m² = 10⁴ cm² | \(0.75 \text{ m}^2 \times \frac{10^4 \text{ cm}^2}{1 \text{ m}^2} = 7.5 \times 10^3 \text{ cm}^2\) |
| 4.6 m² to mm² | 1 m² = 10⁶ mm² | \(4.6 \text{ m}^2 \times \frac{10^6 \text{ mm}^2}{1 \text{ m}^2} = 4.6 \times 10^6 \text{ mm}^2\) |
| 2.8 km² to cm² | 1 km² = 10⁶ m², 1 m² = 10⁴ cm² | \(2.8 \text{ km}^2 \times \frac{10^6 \text{ m}^2}{1 \text{ km}^2} \times \frac{10^4 \text{ cm}^2}{1 \text{ m}^2} = 2.8 \times 10^{10} \text{ cm}^2\) |
| 0.35 hm² to mm² | 1 hm² = 10⁴ m², 1 m² = 10⁶ mm² | \(0.35 \text{ hm}^2 \times \frac{10^4 \text{ m}^2}{1 \text{ hm}^2} \times \frac{10^6 \text{ mm}^2}{1 \text{ m}^2} = 3.5 \times 10^9 \text{ mm}^2\) |
| 120 dam² to dm² | 1 dam² = 10² m², 1 m² = 10² dm² | \(120 \text{ dam}^2 \times \frac{10^2 \text{ m}^2}{1 \text{ dam}^2} \times \frac{10^2 \text{ dm}^2}{1 \text{ m}^2} = 1.2 \times 10^6 \text{ dm}^2\) |
Solved Examples:
Example 1: Convert 0.5 km² to cm²
Method 1: Direct conversion
1 km = 1,000 m = 100,000 cm
So 1 km² = (100,000 cm)² = 10,000,000,000 cm² = 10¹⁰ cm²
\[0.5 \text{ km}^2 \times \frac{10^{10} \text{ cm}^2}{1 \text{ km}^2} = 5 \times 10^9 \text{ cm}^2\]
> Method 2: Step-by-step >
\[0.5 \text{ km}^2 \times \frac{10^6 \text{ m}^2}{1 \text{ km}^2} \times \frac{10^4 \text{ cm}^2}{1 \text{ m}^2} = 0.5 \times 10^{10} \text{ cm}^2 = 5 \times 10^9 \text{ cm}^2\]
> Result: 0.5 km² = 5 × 10⁹ cm² = 5,000,000,000 cm²
Example 2: A rectangular swimming pool is 25 m long and 12.5 m wide. Calculate its area in hectares and square feet.
Step 1: Calculate area in m²
\[\text{Area} = 25 \text{ m} \times 12.5 \text{ m} = 312.5 \text{ m}^2\]
> Step 2: Convert to hectares >
\[312.5 \text{ m}^2 \times \frac{1 \text{ ha}}{10,000 \text{ m}^2} = 0.03125 \text{ ha}\]
> Step 3: Convert to square feet > 1 m² ≈ 10.76 ft² >
\[312.5 \text{ m}^2 \times \frac{10.76 \text{ ft}^2}{1 \text{ m}^2} ≈ 3,363 \text{ ft}^2\]
> Result: Pool area = 312.5 m² = 0.03125 ha ≈ 3,363 ft²
Example 3: A garden plot is 15 m long and 800 cm wide. What is its area in dm²?
Method 1: Convert everything to dm first
Length: 15 m = 150 dm
Width: 800 cm = 80 dm
\[\text{Area} = 150 \text{ dm} \times 80 \text{ dm} = 12,000 \text{ dm}^2\]
> Method 2: Calculate in m², then convert > Width: 800 cm = 8 m >
\[\text{Area} = 15 \text{ m} \times 8 \text{ m} = 120 \text{ m}^2\]
>
\[120 \text{ m}^2 \times \frac{100 \text{ dm}^2}{1 \text{ m}^2} = 12,000 \text{ dm}^2\]
> Result: Garden area = 12,000 dm²
Example 4: A circular field has a diameter of 200 m. Find its area in hectares and acres.
Step 1: Find radius and calculate area
Radius = 200 m ÷ 2 = 100 m
\[A = \pi r^2 = 3.14159 \times (100 \text{ m})^2 = 31,416 \text{ m}^2\]
> Step 2: Convert to hectares >
\[31,416 \text{ m}^2 \times \frac{1 \text{ ha}}{10,000 \text{ m}^2} = 3.14 \text{ ha}\]
> Step 3: Convert to acres >
\[31,416 \text{ m}^2 \times \frac{1 \text{ acre}}{4,047 \text{ m}^2} ≈ 7.76 \text{ acres}\]
> Result: Circular field area = 31,416 m² = 3.14 ha ≈ 7.76 acres
Example 5: A compound shape consists of a rectangle (50 cm × 30 cm) with a semicircle (radius 15 cm) attached to one side. Find the total area in mm².
Step 1: Calculate rectangle area
\[A_{rectangle} = 50 \text{ cm} \times 30 \text{ cm} = 1,500 \text{ cm}^2\]
> Step 2: Calculate semicircle area >
\[A_{semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times (15 \text{ cm})^2 = \frac{1}{2} \times 3.14 \times 225 = 353.25 \text{ cm}^2\]
> Step 3: Calculate total area >
\[A_{total} = 1,500 + 353.25 = 1,853.25 \text{ cm}^2\]
> Step 4: Convert to mm² >
\[1,853.25 \text{ cm}^2 \times \frac{100 \text{ mm}^2}{1 \text{ cm}^2} = 185,325 \text{ mm}^2\]
> Result: Total area = 185,325 mm²