What Is Area?

Area is the amount of two-dimensional space inside a shape.

Imagine covering the floor of a room with tiles. The amount of floor covered by the tiles represents the area of the floor.

Area answers questions such as:

• How much floor needs to be covered with carpet?
• How much paint is needed for a wall?
• How much grass covers a garden?
• How large is a tabletop?
• How much material is needed to cover a surface?

Area measures the space inside a boundary.

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Understanding Square Units

Area is measured using square units.

Common units include:

• mm² — square millimetres
• cm² — square centimetres
• m² — square metres
• km² — square kilometres

A square centimetre, written cm², is the area of a square measuring:

1 cm × 1 cm

A square metre, written m², is the area of a square measuring:

1 m × 1 m

This is why area uses squared units.

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Counting Squares

One way to understand area is to cover a shape with equal-sized squares.

Suppose a rectangle contains:

4 rows

with:

6 squares in each row

The number of squares is:

4 × 6 = 24

Therefore:

Area = 24 square units

This gives us the rule for finding the area of a rectangle.

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Area of a Rectangle

A rectangle has a length and a width.

The area is:

Area = length × width

or:

A = l × w

where:

A = area

l = length

w = width

Worked Example

A rectangle has:

length = 8 cm

width = 5 cm

Therefore:

A = l × w

A = 8 × 5

A = 40 cm²

The area is 40 cm².

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Why Length × Width Works

Suppose a rectangle measures:

7 cm × 3 cm

You could cover it with 1 cm² squares.

There would be:

7 squares across

and:

3 rows

Therefore:

7 × 3 = 21 squares

So:

A = 21 cm²

The formula length × width is a quick way of counting the number of square units inside the rectangle.

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Area of a Square

A square has four equal sides.

Because its length and width are equal:

Area = side × side

or:

A = s²

where:

s = side length

Worked Example

A square has a side length of:

6 cm

Therefore:

A = 6 × 6

A = 36 cm²

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Squaring a Number

The notation:

6²

means:

6 × 6

Therefore:

6² = 36

Similarly:

8² = 8 × 8 = 64

and:

12² = 12 × 12 = 144

So the area formula for a square:

A = s²

means:

area = side × side

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Rectangle Example

A classroom floor measures:

9 m × 7 m

Find its area.

A = l × w

A = 9 × 7

A = 63 m²

The classroom floor has an area of:

63 m²

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Square Example

A square garden has sides measuring:

12 m

Find its area.

A = s²

A = 12²

A = 144 m²

The garden has an area of:

144 m²

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Decimal Measurements

Side lengths do not need to be whole numbers.

Suppose a rectangular table measures:

1.5 m × 0.8 m

Then:

A = 1.5 × 0.8

A = 1.2 m²

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Finding a Missing Length

Sometimes the area and one dimension are known.

Because:

A = l × w

we can rearrange:

l = A ÷ w

or:

w = A ÷ l

Worked Example

A rectangle has an area of:

48 cm²

Its length is:

8 cm

Find its width.

w = A ÷ l

w = 48 ÷ 8

w = 6 cm

Check:

8 × 6 = 48 cm²

Correct.

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Finding the Side of a Square

Suppose a square has an area of:

81 cm²

We need a number that multiplied by itself gives 81.

9 × 9 = 81

Therefore:

side = 9 cm

This can also be written using a square root:

s = √81

s = 9 cm

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Choosing Appropriate Area Units

The unit should suit the size of the object.

Small objects

Use:

mm² or cm²

Examples:

• Postage stamp.
• Phone screen.
• Photograph.

Medium-sized surfaces

Use:

m²

Examples:

• Classroom floor.
• Wall.
• Garden.
• Apartment.

Very large areas

Use:

km²

Examples:

• Cities.
• Lakes.
• Forests.
• Countries.

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Units Must Match

Before calculating area, make sure both measurements use the same unit.

Suppose a rectangle measures:

2 m × 50 cm

Do not calculate:

2 × 50 = 100 m²

The units are different.

Convert first.

50 cm = 0.5 m

Now:

A = 2 × 0.5

A = 1 m²

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Converting Square Units

Area conversions require special care.

Remember:

1 m = 100 cm

But:

1 m² ≠ 100 cm²

A square measuring:

1 m × 1 m

is:

100 cm × 100 cm

Therefore:

1 m² = 10,000 cm²

because:

100 × 100 = 10,000

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Perimeter and Area

Perimeter and area describe different properties.

Perimeter measures the distance around a shape.

Area measures the space inside a shape.

For a rectangle:

Perimeter = 2(l + w)

Area = l × w

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Worked Example: Area and Perimeter

A rectangle measures:

10 cm × 4 cm

Area

A = 10 × 4

A = 40 cm²

Perimeter

P = 2(10 + 4)

P = 28 cm

Notice:

Area = 40 cm²

Perimeter = 28 cm

They have different meanings and different units.

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Practical Example: Flooring

A bedroom measures:

5 m × 4 m

How much flooring is required?

A = 5 × 4

A = 20 m²

Therefore, at least 20 m² of flooring is required.

In a real project, additional material might be purchased to allow for cutting and waste.

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Practical Example: Carpet Cost

A room measures:

6 m × 4 m

Carpet costs:

$25 per m²

First calculate the area:

A = 6 × 4

A = 24 m²

Now calculate the cost:

24 × $25 = $600

Therefore, the carpet costs:

$600

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Practical Example: Painting a Wall

A rectangular wall measures:

5 m wide

and:

2.8 m high

Area:

A = 5 × 2.8

A = 14 m²

If one litre of paint covers:

7 m²

then:

14 ÷ 7 = 2

Approximately 2 litres of paint are required for one coat.

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Practical Example: Grass

A rectangular lawn measures:

15 m × 8 m

Area:

A = 15 × 8

A = 120 m²

If grass seed covers:

20 m² per bag

then:

120 ÷ 20 = 6

The gardener needs:

6 bags of grass seed

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Practical Example: Tiles

A bathroom floor measures:

4 m × 3 m

Area:

A = 4 × 3

A = 12 m²

Suppose each box of tiles covers:

2 m²

Number of boxes:

12 ÷ 2 = 6

Therefore, at least:

6 boxes

are required.

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Practical Example: Square Patio

A square patio has sides of:

4.5 m

Area:

A = 4.5 × 4.5

A = 20.25 m²

If paving costs:

$40 per m²

then:

20.25 × 40 = $810

The paving would cost:

$810

before considering additional costs or waste.

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Compound Rectangular Shapes

Some shapes can be divided into two or more rectangles.

To find the total area:

1. Divide the shape into rectangles.
2. Find the area of each rectangle.
3. Add the areas.

For example:

Area of rectangle A = 24 cm²

Area of rectangle B = 15 cm²

Therefore:

Total area = 24 + 15

Total area = 39 cm²

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Subtracting Areas

Another method is to imagine a complete rectangle and subtract a missing section.

Suppose:

large rectangle area = 80 cm²

and:

missing rectangle area = 20 cm²

Then:

Area of shape = 80 − 20

Area = 60 cm²

Both methods can produce the same result.

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Same Area, Different Shapes

Different rectangles can have the same area.

For example:

Rectangle A

12 cm × 2 cm

A = 24 cm²

Rectangle B

8 cm × 3 cm

A = 24 cm²

Rectangle C

6 cm × 4 cm

A = 24 cm²

All three rectangles have the same area but different dimensions.

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Same Perimeter, Different Areas

Consider:

Rectangle A

8 cm × 2 cm

Perimeter:

20 cm

Area:

16 cm²

Square B

5 cm × 5 cm

Perimeter:

20 cm

Area:

25 cm²

The shapes have the same perimeter but different areas.

This demonstrates why perimeter and area should not be confused.

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Estimating Area

Estimation is useful for checking whether an answer is reasonable.

Suppose a rectangular floor measures:

7.9 m × 5.1 m

Estimate:

8 × 5 = 40 m²

Calculate:

7.9 × 5.1 = 40.29 m²

Since the exact answer is close to the estimate, it appears reasonable.

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Area from a Diagram

When solving an area problem from a diagram:

1. Identify the shape.
2. Identify the required dimensions.
3. Check the units.
4. Choose the appropriate formula.
5. Substitute the values.
6. Calculate.
7. Write the answer using square units.

For a rectangle:

A = l × w

For a square:

A = s²

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Real-World Problem: Garden

A rectangular garden measures:

18 m × 12 m

A gardener wants to cover the entire garden with soil.

Area:

A = 18 × 12

A = 216 m²

If one bag of soil covers:

6 m²

then:

216 ÷ 6 = 36

The gardener needs:

36 bags

assuming the stated coverage is exact and no additional material is required.

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Real-World Problem: Classroom Floor

A classroom measures:

8.5 m × 7 m

Area:

A = 8.5 × 7

A = 59.5 m²

Suppose floor tiles cost:

$30 per m²

Cost:

59.5 × 30

= $1785

Therefore, the basic tile cost is:

$1785

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Real-World Problem: Missing Dimension

A rectangular room has an area of:

72 m²

Its length is:

9 m

Find its width.

A = l × w

72 = 9 × w

w = 72 ÷ 9

w = 8 m

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Real-World Problem: Square Garden

A square garden has an area of:

196 m²

Find the side length.

We need a number that multiplied by itself equals 196.

14 × 14 = 196

Therefore:

side = 14 m

Its perimeter would be:

P = 4 × 14

P = 56 m

This problem connects area and perimeter.

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Common Mistakes

Confusing Area with Perimeter

Remember:

Area = inside

Perimeter = around

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Adding Instead of Multiplying

For a rectangle measuring:

8 cm × 5 cm

Incorrect area:

8 + 5 = 13 cm²

Correct:

8 × 5 = 40 cm²

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Forgetting Square Units

Incorrect:

Area = 40 cm

Correct:

Area = 40 cm²

Area requires square units.

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Squaring Both Dimensions

For a rectangle:

A = l × w

Do not calculate:

l² × w²

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Multiplying Measurements with Different Units

Do not multiply metres by centimetres without first converting them to compatible units.

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Converting Area Like Length

Remember:

1 m = 100 cm

but:

1 m² = 10,000 cm²

Square units require the conversion factor to be squared.

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Assuming Same Perimeter Means Same Area

Two shapes can have equal perimeters but very different areas.

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Using Every Number in a Word Problem

Some problems contain information that is not needed.

Identify what the question actually asks before calculating.

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Check Your Understanding

1. Define area.

2. Why is area measured in square units?

3. What does 1 cm² represent?

4. Find the area of a rectangle measuring 8 cm × 6 cm.

5. Find the area of a rectangle measuring 15 m × 9 m.

6. Find the area of a square with side length 7 cm.

7. Find the area of a square with side length 12 m.

8. A rectangle has an area of 60 cm² and a length of 10 cm. Find its width.

9. A rectangle has an area of 96 m² and a width of 8 m. Find its length.

10. A square has an area of 64 cm². Find its side length.

11. A bedroom measures 5 m × 3.5 m. Find its floor area.

12. Carpet costs $20 per m². Find the cost of carpeting the room in Question 11.

13. A wall measures 6 m × 2.5 m. Find its area.

14. One litre of paint covers 5 m². How many litres are required for one coat on the wall in Question 13?

15. A rectangular garden measures 20 m × 14 m. Find its area.

16. A square playground has sides measuring 25 m. Find its area.

17. A rectangle measures 2 m × 60 cm. Find its area in m².

18. Explain why 1 m² equals 10,000 cm² rather than 100 cm².

19. A rectangle measures 10 cm × 4 cm. Calculate both its area and perimeter.

20. Explain the difference between area and perimeter.

21. Give an example of two rectangles with an area of 24 cm² but different dimensions.

22. A rectangular floor has an area of 84 m² and a width of 7 m. Find its length.

23. A square has an area of 225 cm². Find its side length.

24. A rectangular lawn measures 12 m × 9 m. Grass seed covers 18 m² per bag. How many bags are required?

25. A square patio has sides of 6 m. Paving costs $35 per m². Find the total basic paving cost.

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Key Terms

• Area – amount of two-dimensional space inside a shape.
• Square unit – unit used to measure area.
• Square centimetre (cm²) – area of a square measuring 1 cm × 1 cm.
• Square metre (m²) – area of a square measuring 1 m × 1 m.
• Rectangle – quadrilateral with four right angles and equal opposite sides.
• Square – quadrilateral with four equal sides and four right angles.
• Length – measurement of one dimension of a rectangle.
• Width – measurement perpendicular to the length.
• Compound shape – shape made from two or more simpler shapes.
• Perimeter – total distance around a two-dimensional shape.

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Key Takeaways

• Area measures the amount of space inside a two-dimensional shape.
• Area is measured in square units such as mm², cm², m², and km².
• A square unit represents a 1 unit × 1 unit square.
• Rectangle: A = l × w
• Square: A = s²
• A missing dimension can be found by dividing the area by the known dimension.
• Both dimensions must use compatible units before calculating area.
• Area-unit conversions are different from length conversions.
• 1 m² = 10,000 cm²
• Compound rectangular shapes can be divided into smaller rectangles.
• Practical area problems include flooring, painting, gardening, tiling, and construction.
• Perimeter measures the distance around a shape.
• Area measures the space inside a shape.
• Perimeter uses linear units such as m.
• Area uses square units such as m².
• Different shapes can have the same area but different dimensions and perimeters.

A simple way to remember the main formulas is:

Rectangle: Area = length × width

Square: Area = side × side