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Year 6 Maths Length, Perimeter & Area

Area of rectangles

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Theory

Area is the amount of surface a shape covers, counted in square units. A rectangle \(7\) cm long and \(5\) cm wide is covered by \(5\) rows of \(7\) one-centimetre squares, so its area is \(7 \times 5 = 35\,\text{cm}^2\). This Year 6 maths page shows how to use area = length \(\times\) width, choose cm² or m², find a missing side, and solve area problems.

Area is measured in square units. A square \(1\) cm by \(1\) cm has an area of \(1\) square centimetre (\(1\,\text{cm}^2\)); a square \(1\) m by \(1\) m has an area of \(1\,\text{m}^2\).

For a rectangle, area = length \(\times\) width: the number of rows times the number of squares in each row.

A square has equal sides, so its area is side \(\times\) side: a \(9\) cm square has area \(81\,\text{cm}^2\).

If you know the area and one side, divide to find the other side: \(56 \div 8 = 7\).

Area is not perimeter. Perimeter is the distance around the outside (add the sides); area is the space inside (multiply).

A 6 by 3 rectangle on a grid A rectangle 6 squares long and 3 squares wide, covered by 18 shaded one-centimetre squares.
\(3\) rows of \(6\) squares: \(6 \times 3 = 18\,\text{cm}^2\).
Rectangle 11 m by 6 m A rectangle labelled 11 m long and 6 m wide. 11 m 6 m
Area \(= 11 \times 6 = 66\,\text{m}^2\).
Rectangle with a missing side A rectangle with the long side labelled 8 cm and the short side labelled with a question mark. Its area is 56 square centimetres. 8 cm ? 56 cm²
Area \(56\,\text{cm}^2\) and length \(8\) cm: the width is \(56 \div 8 = 7\) cm.

Area of rectangles and squares:

\[ A = \text{length} \times \text{width} \]
A = length × width
\[ A_{\text{square}} = \text{side} \times \text{side} \]
A of a square = side × side
\[ \text{width} = A \div \text{length} \]
width = A ÷ length
Square units. Write area in \(\text{cm}^2\) or \(\text{m}^2\), not cm or m.
Choose the unit. Use \(\text{cm}^2\) for small things like a book cover, \(\text{m}^2\) for rooms and gardens.

Finding the area

  1. Check both sides are in the same unit.
  2. Multiply length by width.
  3. Write square units.

Finding a missing side

  1. Divide the area by the side you know.
Example 1 — Count and multiply
Use the first diagram. Find the area by counting and by multiplying.
Solution
\(6 \times 3\)\(=\)\(18\)
6×3=18

\(18\) squares \(= 18\,\text{cm}^2\).

Example 2 — A rectangle
Find the area of a rectangle \(7\) cm long and \(5\) cm wide.
Solution
\(7 \times 5\)\(=\)\(35\)
7×5=35

\(35\,\text{cm}^2\).

Example 3 — A square
A square card has sides of \(9\) cm. Find its area.
Solution
\(9 \times 9\)\(=\)\(81\)
9×9=81

\(81\,\text{cm}^2\).

Example 4 — A floor
A floor is \(11\) m long and \(6\) m wide. Find its area.
Solution
\(11 \times 6\)\(=\)\(66\)
11×6=66

\(66\,\text{m}^2\), as in the second diagram.

Example 5 — Missing side
(i) A rectangle has area \(56\,\text{cm}^2\) and length \(8\) cm. Find its width.
(ii) A square rug has area \(64\,\text{m}^2\). How long is each side?
Solution
\(56 \div 8\)\(=\)\(7\)
\(8 \times 8\)\(=\)\(64\)
56÷8=7,8×8=64

(i) \(7\) cm
(ii) \(8\) m

Example 6 — Mats
A gym floor is \(10\) m by \(6\) m. Mats are \(2\) m by \(1\) m.
(i) What is the area of the floor?
(ii) How many mats cover it?
Solution
\(10 \times 6\)\(=\)\(60\)
\(60 \div 2\)\(=\)\(30\)
60 m ,30 mats

(i) \(60\,\text{m}^2\)
(ii) \(30\) mats (each covers \(2\,\text{m}^2\))

Example 7 — Same area, different perimeter
Compare a \(10\) m by \(3\) m rectangle with a \(6\) m by \(5\) m rectangle.
Solution
\(10 \times 3\)\(=\)\(6 \times 5\)
\(2 \times (10 + 3)\)\(=\)\(26\)
\(2 \times (6 + 5)\)\(=\)\(22\)
both 30 m , perimeters 26 m and 22 m

Both have area \(30\,\text{m}^2\), but the perimeters are \(26\) m and \(22\) m.

Example 8 — Find the mistake
For a rectangle \(9\) m by \(5\) m, Tom added \(9 + 5 + 9 + 5\) and said the area is \(28\,\text{m}^2\). What is right?
Solution
\(9 \times 5\)\(=\)\(45\)
9×5=45

He found the perimeter (\(28\) m). The area is \(9 \times 5 = 45\,\text{m}^2\).

Common pitfalls

Adding instead of multiplying. Adding the sides gives the perimeter, not the area.
Forgetting square units. Area is in \(\text{cm}^2\) or \(\text{m}^2\).
Doubling a square. A \(9\) cm square has area \(81\,\text{cm}^2\), not \(18\).
Mixing units. Change both sides to the same unit before multiplying.

Frequently asked questions

How do you find the area of a rectangle?

Multiply the length by the width.

What is a square centimetre?

The area of a square \(1\) cm by \(1\) cm, written \(1\,\text{cm}^2\).

How do you find a missing side?

Divide the area by the known side: \(56 \div 8 = 7\).

Can two rectangles have the same area but different perimeters?

Yes: \(10\) by \(3\) and \(6\) by \(5\) both have area \(30\).

What happens to the area if both the length and the width double?

It becomes \(4\) times as big.

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