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 UnitArea FactorExample Conversion
Mm²(10²)⁶ = 100¹²1 Mm² = 10²⁴ m²
km²(10²)³ = 100³1 km² = 10⁶ m²
hm²(10²)² = 100²1 hm² = 10⁴ m²
dam²(10²)¹ = 100¹1 dam² = 10² m²
m²(10²)⁰ = 100⁰Base unit
dm²(10²)⁻¹ = 100⁻¹1 m² = 10⁻² dm²
cm²(10²)⁻² = 10⁻⁴1 cm² = 10⁻⁴ m²
mm²(10²)⁻³ = 100⁻³1 mm² = 10⁻⁶ m²
μm²(10²)⁻⁶ = 100⁻¹²1 μm² = 10⁻²⁴ m²
Desired ConversionEquivalence RelationshipFormula and Calculation
2.5 km² to m²1 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 m²1 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 m²1 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 dm²1 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 cm²1 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 mm²1 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 cm²1 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 mm²1 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 dm²1 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

1. Convert 3.4 cm² to mm²: (1 points)

2. Convert 2.5 m² to cm²: (1 points)

3. Convert 500 mm² to cm²: (1 points)

4. Convert 0.75 ha to m² (1 ha = 10 000 m²): (1 points)

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

a ) Convert 5 m² to cm²:

b ) Convert 2.4 cm² to mm²:

c ) Convert 0.3 km² to m²:

d ) Convert 1 500 mm² to cm²:

e ) Convert 0.75 ha to m²:

f ) Convert 45 dm² to cm²:

g ) Convert 0.8 m² to mm²:

h ) Convert 250 cm² to m²:

6. Mixed area unit conversions: (12 points)

a ) Convert 2 acres to m² (1 acre = 4 047 m²):

b ) Convert 1 square foot to cm² (1 ft = 30.48 cm):

c ) Convert 0.5 km² to acres (1 acre = 4 047 m²):

d ) Convert 25 square yards to m² (1 yd = 0.914 m):

7. Geometric area calculations: (15 points)

a ) A square has sides of 50 cm. What is its area in m²?

b ) A rectangle is 2.5 m × 80 cm. What is its area in cm²?

c ) A circle has radius 10 cm. What is its area in mm² (use π ≈ 3.14)?

d ) A triangle has base 60 mm and height 40 mm. What is its area in cm²?

e ) A parallelogram has base 15 cm and height 8 cm. What is its area in mm²?