Practice Maths

L22 — Calculating Area of Rectangles

Key Terms

area
The amount of flat surface enclosed inside a 2D shape. Measured in square units (cm², m², etc.).
perimeter
The total distance around the outside of a shape. Measured in linear units (cm, m, etc.).
square units
Units used for area: mm², cm², m², km². A square unit is a 1×1 square of that unit.
composite shape
A shape made by joining two or more simple shapes together.
missing dimension
An unknown side length found by rearranging the area formula: l = A ÷ w.

Area of a Rectangle or Square

A = l × w

where l is the length and w is the width. For a square, l = w = s, so A = s2.

Square Units: Area is always in square units because you multiply length × length. cm × cm = cm². Never write just “cm” for an area answer.

Worked Example — Rectangle

Find the area and perimeter of a 9 cm × 5 cm rectangle.

Step 1 — Area: A = l × w = 9 × 5 = 45 cm².

Step 2 — Perimeter: P = 2(l + w) = 2(9 + 5) = 2 × 14 = 28 cm.

Worked Example — Composite Shape

Find the total area of an L-shaped room: outer dimensions 8 m × 6 m, with a 3 m × 2 m corner removed.

Step 1 — Area of full rectangle: 8 × 6 = 48 m².

Step 2 — Area of removed corner: 3 × 2 = 6 m².

Step 3 — Remaining area: 48 − 6 = 42 m².

Finding a Missing Dimension

Rearrange A = l × w:

  • Missing length: l = A ÷ w
  • Missing width: w = A ÷ l
Area vs Perimeter: Area is the space inside (cm²). Perimeter is the distance around the outside (cm). They use different formulas and different units — never mix them up.

What Is Area?

Area is the amount of flat surface inside a 2D shape. Think of it as how many 1×1 squares fit inside. A 4 cm × 3 cm rectangle fits exactly 12 unit squares inside — so its area is 12 cm².

We measure area in square units: mm², cm², m², or km². The unit depends on the context — cm² for small objects, m² for rooms, km² for large areas.

A square metre (m²) is a 1 m × 1 m square. There are 10,000 cm² in every 1 m² (because 100 cm × 100 cm = 10,000 cm²). This is a surprisingly large number — a common mistake is treating 1 m² as if it equals 100 cm².

Area of a Rectangle

The formula is A = l × w. This always produces a square-unit answer. A square is a special rectangle where l = w, so its area formula becomes A = s² (side squared).

Perimeter of a Rectangle

Perimeter is the total distance around the outside. The formula is P = 2(l + w), or equivalently P = 2l + 2w. Note that perimeter is in cm (or m), not cm². It is easy to confuse area and perimeter, especially in word problems — always check what the question is asking for.

Finding a Missing Dimension

If you know the area and one side, rearrange the formula:

  • A = l × w, so if A = 48 and w = 6, then l = 48 ÷ 6 = 8.
Always check your answer by substituting back: does l × w equal the given area? If not, revisit your working.

Composite Shapes

A composite shape is made by joining or cutting simple shapes. Two strategies:

  • Additive: Split the composite shape into two rectangles. Find each area separately and add them.
  • Subtractive: Find the area of the full outer rectangle, then subtract the area of the cut-out portion.
Always label all dimensions before calculating. For L-shapes and similar figures, some dimensions may need to be calculated from the given ones before you can use the area formula.

  1. Selecting the Right Unit

    Choose the most appropriate unit for each area measurement: mm², cm², m², or km².

    1. A school’s rugby oval.
    2. The screen of a smartphone.
    3. The state of Queensland.
    4. The wing of a small butterfly.
    5. A square has an area of 1 m². How many cm² is this? (Recall: 1 m = 100 cm.)

  2. Basic Area Calculation

    Calculate the area of each rectangle. Include the correct unit in your answer.

    12 cm 4 cm (a)
    9 m 7 m (b)
    15 mm 2 mm (c)
    8 cm 8 cm (d)
    25 m 3 m (e)
    1. Find the area of rectangle (a).
    2. Find the area of rectangle (b).
    3. Find the area of rectangle (c).
    4. Find the area of square (d). How does the formula change compared to a non-square rectangle?
    5. Find the area of rectangle (e). What real-world object could this represent?

  3. Area of Squares

    A square is a rectangle with all sides equal. Find the area of squares with the following side lengths.

    1. Side = 5 cm
    2. Side = 11 m
    3. Side = 20 km
    4. Side = 0.5 cm
    5. A square has an area of 144 cm². Find the side length. (Hint: what number multiplied by itself gives 144?)

  4. Working with Decimals

    Find the area of each rectangle. Show your working.

    6.4 m 3 m (a)
    4.5 cm 2.2 cm (b)
    7.5 m 4 m (c)
    3.2 cm 5.5 cm (d)
    1. Find the area of rectangle (a).
    2. Find the area of rectangle (b). Round to 2 decimal places.
    3. Find the area of rectangle (c).
    4. Find the area of rectangle (d). Round to 2 decimal places.
    5. A floor tile is 0.6 m × 0.4 m. Find its area in m², then convert to cm².

  5. Finding the Missing Dimension

    The area and one dimension are given for each rectangle. Find the missing dimension.

    A = 48 cm² 6 cm l = ? (a)
    A = 100 m² 25 m w = ? (b)
    A = 36 cm² s = ? s = ? (c)
    A = 5.6 m² 2 m l = ? (d)
    1. Find the missing length in diagram (a). Show the rearrangement of the formula.
    2. Find the missing width in diagram (b).
    3. In diagram (c), the shape is a square. Find the side length.
    4. Find the missing length in diagram (d).

  6. Compound Shape — Additive

    Calculate the total area by splitting the L-shape into two rectangles.

    4 cm 10 cm 12 cm 6 cm
    1. What are the two unlabeled dimensions of this shape? Calculate them using the given measurements.
    2. Describe how you would split this L-shape into two rectangles. Give the dimensions of each.
    3. Find the area of each of your two rectangles.
    4. Find the total area of the L-shape.

  7. Compound Shape — Subtractive

    The shaded region shows the remaining area after a rectangular piece is cut from the corner.

    16 m 10 m cut-out 6 m × 4 m shaded area
    1. Find the area of the full outer rectangle (before the cut-out).
    2. Find the area of the cut-out (dashed rectangle).
    3. Find the shaded (remaining) area.
    4. If the cut-out were doubled in size to 12 m × 8 m, what would the new remaining area be?

  8. Aussie Backyard

    A family is laying turf in their backyard. The rectangular yard is 15 m long and 8 m wide. In one corner, there is a paved area measuring 3 m by 4 m that will not be turfed.

    Lawn Paved 15 m 8 m 3 m 4 m
    1. Find the total area of the yard.
    2. Find the area of the paved section.
    3. Find the area that will be covered in turf.
    4. Turf costs $12 per m². Find the total cost to turf the grass area.

  9. Error Analysis

    Henry calculated the area of a 5 cm × 8 cm rectangle as 26 cm2.

    1. What mistake did Henry most likely make? Identify the formula he used.
    2. What is the correct area? Show your working.
    3. What would the perimeter of this rectangle be? Show that Henry’s incorrect answer was actually the perimeter, not the area.

  10. The Variable Rectangle

    A rectangle has a length of 10 cm and an area of 70 cm².

    1. Find the width of the rectangle.
    2. If the width is doubled, what is the new area? What does this tell you about the relationship between one dimension and area?
    3. A different rectangle also has an area of 70 cm², but its width is 5 cm. Find its length and perimeter. Compare the perimeters of the two rectangles.
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