Area of Rectangle
The area of a rectangle is the number of unit squares that cover it, found quickly as \(\text{length} \times \text{width}\). In NSW Year 7 maths (NESA outcome MA4‑ARE‑C‑01) this is the first area formula, and the formulas for triangles and parallelograms grow out of it.
You will count squares in rows, use \(A = l \times w\) and \(A = s \times s\) for a square, find a missing side by dividing the area by the known side, change both sides to the same unit before multiplying, cost jobs such as painting a stage at a price per square metre, and compare rectangles made from the same 36 unit squares. It is part of Year 7 Area and Volume.
Every question with a fully worked solution.
- Area of Rectangle - Video - Area of a Rectangle Watch
Theory
The area of a shape is the amount of flat surface it covers, measured in square units. For a rectangle, \(A = l \times w\) (length times width, also written \(A = L \times B\) with \(B\) for breadth). This Year 7 maths page shows where the formula comes from, how to find a missing side and how to work with mixed units.
A square centimetre (\(1~\text{cm}^2\)) is a square \(1\) cm by \(1\) cm. The area is the number of these squares that cover the shape.
Count in rows. A rectangle \(5\) cm long and \(3\) cm wide holds \(3\) rows of \(5\) squares: \(3 \times 5 = 15~\text{cm}^2\). The shortcut is the formula \(A = l \times w\). You may also see \(A = L \times B\) or \(A = LB\), where \(B\) is the breadth (another word for width).
Square: length and width are equal, so \(A = s \times s\).
Missing side: divide the area by the side you know. If \(A = 56~\text{cm}^2\) and \(l = 8\) cm, then \(w = 56 \div 8 = 7\) cm. For a square, find the number that times itself gives the area.
Units: the sides must be in the same unit before you multiply. Write the answer in square units.
Formulas:
| Rectangle | Area | Perimeter |
|---|---|---|
| \(12\) by \(1\) | \(12~\text{cm}^2\) | \(26\) cm |
| \(6\) by \(2\) | \(12~\text{cm}^2\) | \(16\) cm |
| \(4\) by \(3\) | \(12~\text{cm}^2\) | \(14\) cm |
How to find the area of a rectangle
- Change both sides to the same unit.
- Multiply length by width.
- Write the answer in square units.
How to find a missing side
- Substitute into \(A = l \times w\).
- Divide the area by the known side.
- Check by multiplying back.
(i) a rectangle \(14\) cm long and \(9\) cm wide
(ii) a square sandpit with sides of \(2.5\) m
| \(14 \times 9\) | \(=\) | \(126\) |
| \(2.5 \times 2.5\) | \(=\) | \(6.25\) |
(i) \(126~\text{cm}^2\)
(ii) \(6.25~\text{m}^2\)
| \(240 \div 100\) | \(=\) | \(2.4\) |
| \(5 \times 2.4\) | \(=\) | \(12\) |
\(12~\text{m}^2\)
(ii) A square chessboard has an area of \(1600~\text{cm}^2\). How long is each side?
| \(91 \div 13\) | \(=\) | \(7\) |
| \(40 \times 40\) | \(=\) | \(1600\) |
(i) \(7\) m (check: \(13 \times 7 = 91\))
(ii) \(40\) cm
| \(60 \div 100\) | \(=\) | \(0.6\) |
| \(1.8 \times 0.6\) | \(=\) | \(1.08\) |
\(1.08~\text{m}^2\). Check in cm: \(180 \times 60 = 10\,800~\text{cm}^2 = 1.08~\text{m}^2\).
| \(48 \div 4\) | \(=\) | \(12\) |
| \(12 \times 12\) | \(=\) | \(144\) |
| \(144 \times 15\) | \(=\) | \(2160\) |
\(\$2160\)
| \(10 + 2 + 2\) | \(=\) | \(14\) |
| \(4 + 2 + 2\) | \(=\) | \(8\) |
| \(14 \times 8\) | \(=\) | \(112\) |
| \(10 \times 4\) | \(=\) | \(40\) |
| \(112 - 40\) | \(=\) | \(72\) |
The outer rectangle is \(14\) m by \(8\) m. Path \(= 112 - 40 = 72~\text{m}^2\).
| \(26 - 2 \times 8\) | \(=\) | \(10\) |
| \(10 \div 2\) | \(=\) | \(5\) |
| \(8 \times 5\) | \(=\) | \(40\) |
Width \(5\) cm, area \(40~\text{cm}^2\).
Common pitfalls
Frequently asked questions
What is the formula for the area of a rectangle?
\(A = l \times w\) (length times width, or breadth).
How do you find the area of a square?
Multiply the side by itself: \(A = s \times s\).
How do you find the width if you know the area?
Divide the area by the length. \(91 \div 13 = 7\).
Why must the units be the same?
Otherwise the number is wrong: \(2\) m by \(30\) cm is \(0.6~\text{m}^2\), not \(60\).
Which rectangle with a given area has the smallest perimeter?
The one closest to a square.