GCSE Maths: Perimeter and area of rectangles and squares (Higher)
Part of the GCSE Maths perimeter, area and circles chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Perimeter and area of rectangles and squares
Two measurements describe how big a flat shape is, and they are easy to mix up. The perimeter is the distance all the way round the outside edge. Imagine walking round the edge of a football pitch: the distance you walk is the perimeter. The area is the amount of flat surface inside the shape. Imagine painting the pitch: the amount of paint you need depends on the area.
Because perimeter is a distance, we measure it in ordinary units of length: millimetres (mm), centimetres (cm), metres (m) or kilometres (km). Area is measured by counting squares, so it uses square units: cm², m², and so on. A cm² (“one square centimetre”) is a square whose sides are each 1 cm long.
Rectangles
Look at a rectangle 5 cm long and 3 cm wide. We can fill it with 1 cm squares: 3 rows, each with 5 squares. Instead of counting every square we use multiplication: area = length × width. That is 5 × 3 = 15 cm².
To go round the rectangle we add up its four sides. Opposite sides of a rectangle are equal, so the sides are length, width, length, width. The perimeter is 2 × (length + width), or simply add all four sides. Here that is 5 + 3 + 5 + 3 = 16 cm.
Squares
A square is a rectangle whose sides are all equal, so the area is side × side (which is why we say a number is “squared”) and the perimeter is 4 × side.
Working backwards
Sometimes you are given the area or the perimeter and have to find a missing side. Use the same formulas, but undo them. If the area is 54 cm² and one side is 9 cm, then 9 × (missing side) = 54, so the missing side is 54 ÷ 9 = 6 cm. If you know the perimeter, take away the sides you know and share the rest between the sides that are left.
Key ideas
- Perimeter = distance round the outside. Add all the sides. Unit: cm, m, ...
- Area = space inside. Unit: cm², m², ...
- Rectangle: area = length × width; perimeter = 2 × (length + width).
- Square: area = side × side; perimeter = 4 × side.
- To find a missing side from the area, divide the area by the side you know.
Watch out
Perimeter and area are different. Adding the sides gives the perimeter (in cm); multiplying gives the area (in cm²). A rectangle 7 cm by 5 cm has perimeter 24 cm but area 35 cm². Always write the units: they tell you which one you found.Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area counts the unit squares inside the shape. A rectangle has length squares in each row and width rows, so we multiply the two.
- Write the formula. There are length squares in every row and width rows, so the number of squares is length × width.Area = length × width
- Substitute the numbers. Both lengths must be in the same unit before you multiply.Area = 9 × 4
- Multiply. Work out the product.9 × 4 = 36
- Write the units. Each little square is 1 cm by 1 cm, so the area is measured in square cm.Area = 36 cm²
Check. Add the rows instead: 9 + 9 + 9 + 9 = 36. ✓
Watch out. Area is in square units (cm²), not cm. Do not add the sides: adding is for perimeter.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Perimeter is the distance all the way round. A rectangle has two lengths and two widths, so add all four sides.
- Mark the four sides. Opposite sides of a rectangle are equal, so the sides are length, width, length, width.12.5, 8, 12.5, 8
- Add the four sides. The perimeter is the total length of the boundary.12.5 + 8 + 12.5 + 8 = 41
- Write the units. Perimeter is a length, so we use m, not square units.Perimeter = 41 m
Check. Shortcut: 2 × (length + width) = 2 × 20.5 = 41. ✓
Watch out. Perimeter is a length (m). Do not give it in square units.
Method. The perimeter is two lengths plus two widths. Take away the two lengths you know, then share what is left between the two widths.
- Take away the two lengths. A rectangle has two sides of length 11 cm. Whatever is left of the perimeter is the two widths.2 × 11 = 22
38 − 22 = 16 - Share between the two widths. There are two equal widths, so divide by 2.16 ÷ 2 = 8
- Write the answer. Give the unit.Width = 8 cm
Check. Add the four sides: 11 + 8 + 11 + 8 = 38 cm. ✓
Watch out. Half the perimeter is length + width, not just the width. Take away the length too.
Method. Find the width and length from the perimeter, then multiply them for the area.
- Call the width w. The length is three times as long, so it is 3w.width = w, length = 3w
- Write the perimeter. Add the four sides: two widths and two lengths.w + 3w + w + 3w = 8w
8w = 48 - Solve for w. Divide both sides by 8.w = 48 ÷ 8 = 6 cm
- Find the length. Length = 3 × width.3 × 6 = 18 cm
- Find the area. Area = length × width.18 × 6 = 108 cm²
Check. Perimeter check: 2 × (18 + 6) = 2 × 24 = 48 cm. ✓
Watch out. The question asks for the area, not for the width. Read the last line of the question before you finish.
Method. Write the perimeter as an expression, make it equal to 34, and solve the equation.
- Write the perimeter. Add the four sides: two lengths and two widths.P = (x + 5) + x + (x + 5) + x = 4x + 10
- Make an equation. The perimeter is 34 cm.4x + 10 = 34
- Solve it. Subtract 10 from both sides, then divide by 4.4x = 24
x = 6 - Find the sides. Width = x. Length = x + 5.width = 6 cm, length = 6 + 5 = 11 cm
- Find the area. Length × width.11 × 6 = 66 cm²
Check. Perimeter: 2 × (11 + 6) = 34 cm. ✓
Watch out. Collect the like terms carefully: there are four sides, so 4x, not 2x.
Exercise 18A Perimeter and area of rectangles and squares
Non-calculatorKey words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area is the number of unit squares inside the shape, so count them. Counting row by row avoids missing or double-counting any.
- Count the squares in each row. Go along each row from the left, one row at a time.Row 1: 3 squares
Row 2: 1 square
Row 3: 2 squares - Add the rows. The area is the total number of squares.3 + 1 + 2 = 6
- Write the units. Each square is 1 cm by 1 cm, so the area is in square centimetres.Area = 6 cm²
Check. Count again by columns instead and you should get the same total. ✓
Watch out. Area is the space inside (cm²). Do not count the edges.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Perimeter is the distance round the outside edge. Each side of a square is 1 cm, so count the outside edges of squares.
- Count the outside edges facing up and facing down. An edge is on the outside if there is no square on the other side of it.Facing up: 4
Facing down: 4 - Count the outside edges facing left and facing right. Do the same for the sides.Facing left: 3
Facing right: 3 - Add all four counts. Every outside edge is 1 cm long.4 + 4 + 3 + 3 = 14
- Write the units. Perimeter is a length.Perimeter = 14 cm
Check. Trace round the shape with your finger, counting each side of a square once. You should reach the same total. ✓
Watch out. Edges that are inside the shape (between two shaded squares) are not part of the perimeter.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area is the number of unit squares inside the shape, so count them. Counting row by row avoids missing or double-counting any.
- Count the squares in each row. Go along each row from the left, one row at a time.Row 1: 4 squares
Row 2: 4 squares
Row 3: 1 square - Add the rows. The area is the total number of squares.4 + 4 + 1 = 9
- Write the units. Each square is 1 cm by 1 cm, so the area is in square centimetres.Area = 9 cm²
Check. Count again by columns instead and you should get the same total. ✓
Watch out. Area is the space inside (cm²). Do not count the edges.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Perimeter is the distance round the outside edge. Each side of a square is 1 cm, so count the outside edges of squares.
- Count the outside edges facing up and facing down. An edge is on the outside if there is no square on the other side of it.Facing up: 4
Facing down: 4 - Count the outside edges facing left and facing right. Do the same for the sides.Facing left: 3
Facing right: 3 - Add all four counts. Every outside edge is 1 cm long.4 + 4 + 3 + 3 = 14
- Write the units. Perimeter is a length.Perimeter = 14 cm
Check. Trace round the shape with your finger, counting each side of a square once. You should reach the same total. ✓
Watch out. Edges that are inside the shape (between two shaded squares) are not part of the perimeter.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Perimeter is the distance all the way round. A rectangle has two lengths and two widths, so add all four sides.
- Mark the four sides. Opposite sides of a rectangle are equal, so the sides are length, width, length, width.9, 4, 9, 4
- Add the four sides. The perimeter is the total length of the boundary.9 + 4 + 9 + 4 = 26
- Write the units. Perimeter is a length, so we use cm, not square units.Perimeter = 26 cm
Check. Shortcut: 2 × (length + width) = 2 × 13 = 26. ✓
Watch out. Perimeter is a length (cm). Do not give it in square units.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area counts the unit squares inside the shape. A rectangle has length squares in each row and width rows, so we multiply the two.
- Write the formula. There are length squares in every row and width rows, so the number of squares is length × width.Area = length × width
- Substitute the numbers. Both lengths must be in the same unit before you multiply.Area = 9 × 4
- Multiply. Work out the product.9 × 4 = 36
- Write the units. Each little square is 1 cm by 1 cm, so the area is measured in square cm.Area = 36 cm²
Check. Add the rows instead: 9 + 9 + 9 + 9 = 36. ✓
Watch out. Area is in square units (cm²), not cm. Do not add the sides: adding is for perimeter.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. A square has four equal sides, so the perimeter is 4 × the side length.
- Write the four sides. All four sides are equal.7 + 7 + 7 + 7
- Add (or multiply by 4). Four equal sides: repeated addition is multiplication.4 × 7 = 28
- Write the units. Perimeter is a length.Perimeter = 28 cm
Check. Divide back: 28 ÷ 4 = 7, the side we started with. ✓
Watch out. Perimeter is the distance round the edge: 4 × side, not side × side.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. A square is a rectangle whose length and width are the same, so the area is side × side.
- Write the formula. All four sides of a square are equal, so length = width = side.Area = side × side
- Substitute. Put the side length in twice.Area = 7 × 7
- Multiply. Work out the product.7 × 7 = 49
- Write the units. Area is measured in square cm.Area = 49 cm²
Check. Picture 7 rows of 7 squares: 7 × 7 = 49. ✓
Watch out. Side × side means 7 × 7, not 7 × 2.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area counts the unit squares inside the shape. A rectangle has length squares in each row and width rows, so we multiply the two.
- Write the formula. There are length squares in every row and width rows, so the number of squares is length × width.Area = length × width
- Substitute the numbers. Both lengths must be in the same unit before you multiply.Area = 12.5 × 8
- Multiply. Work out the product.12.5 × 8
Ignore the decimal points: 125 × 8 = 1000
Together there are 1 decimal place, so 1000 becomes 100 - Write the units. Each little square is 1 m by 1 m, so the area is measured in square m.Area = 100 m²
Check. Estimate with whole numbers: 12 × 8 = 96, which is close to 100. ✓
Watch out. Area is in square units (m²), not m. Do not add the sides: adding is for perimeter.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Perimeter is the distance all the way round. A rectangle has two lengths and two widths, so add all four sides.
- Mark the four sides. Opposite sides of a rectangle are equal, so the sides are length, width, length, width.12.5, 8, 12.5, 8
- Add the four sides. The perimeter is the total length of the boundary.12.5 + 8 + 12.5 + 8 = 41
- Write the units. Perimeter is a length, so we use m, not square units.Perimeter = 41 m
Check. Shortcut: 2 × (length + width) = 2 × 20.5 = 41. ✓
Watch out. Perimeter is a length (m). Do not give it in square units.
Method. Area = length × width. We know the area and one side, so we undo the multiplication with a division to find the missing width.
- Write the formula and substitute. Put in what we know and use a question mark for the missing side.Area = length × width
54 = 9 × ? - Undo the multiplication. The opposite (inverse) of multiplying by 9 is dividing by 9.? = 54 ÷ 9 = 6
- Write the answer with its unit. A side is a length.The missing side is 6 cm
Check. Multiply back: 9 × 6 = 54 cm². ✓
Watch out. The area is in square units but the missing side is a length, so the answer is in cm, not cm².
Method. A square has four equal sides, so one side is the perimeter divided by 4.
- Divide the perimeter by 4. The perimeter is four equal sides added together.36 ÷ 4 = 9
- Write the answer. Give the unit.Side = 9 cm
Check. 4 × 9 = 36. ✓
Watch out. This is the side length, not the area.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. A square is a rectangle whose length and width are the same, so the area is side × side.
- Write the formula. All four sides of a square are equal, so length = width = side.Area = side × side
- Substitute. Put the side length in twice.Area = 9 × 9
- Multiply. Work out the product.9 × 9 = 81
- Write the units. Area is measured in square cm.Area = 81 cm²
Check. Picture 9 rows of 9 squares: 9 × 9 = 81. ✓
Watch out. Side × side means 9 × 9, not 9 × 2.
Method. The perimeter is two lengths plus two widths. Take away the two lengths you know, then share what is left between the two widths.
- Take away the two lengths. A rectangle has two sides of length 6 cm. Whatever is left of the perimeter is the two widths.2 × 6 = 12
30 − 12 = 18 - Share between the two widths. There are two equal widths, so divide by 2.18 ÷ 2 = 9
- Write the answer. Give the unit.Width = 9 cm
Check. Add the four sides: 6 + 9 + 6 + 9 = 30 cm. ✓
Watch out. Half the perimeter is length + width, not just the width. Take away the length too.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area counts the unit squares inside the shape. A rectangle has length squares in each row and width rows, so we multiply the two.
- Write the formula. There are length squares in every row and width rows, so the number of squares is length × width.Area = length × width
- Substitute the numbers. Both lengths must be in the same unit before you multiply.Area = 9 × 6
- Multiply. Work out the product.9 × 6 = 54
- Write the units. Each little square is 1 cm by 1 cm, so the area is measured in square cm.Area = 54 cm²
Check. Add the rows instead: 9 + 9 + 9 + 9 + 9 + 9 = 54. ✓
Watch out. Area is in square units (cm²), not cm. Do not add the sides: adding is for perimeter.
Method. Find the perimeter (the distance round the field), then multiply by the cost of each metre.
- Find the perimeter. The fence goes all the way round, so we need the perimeter.120 + 85 + 120 + 85 = 410 m
- Multiply by the cost per metre. Each metre costs £6.410 × 6 = £2,460
Check. Estimate: 400 m × £6 = £2,400, close to £2,460. ✓
Watch out. Fencing goes round the edge: use the perimeter, not the area.
Method. Check which measurement Aisha has worked out, then use the correct formula.
- Look at what Aisha did. Adding all four sides gives the distance round the edge. That is the perimeter (24 cm), not the area.7 + 5 + 7 + 5 = 24 cm
- Use the area formula. Area is length × width, which counts squares inside the shape.Area = 7 × 5 = 35 cm²
- Write the conclusion. Say both what was wrong and the right answer.Aisha found the perimeter (24 cm) instead of the area. The area is 35 cm².
Check. Five rows of 7 squares: 7 + 7 + 7 + 7 + 7 = 35. ✓
Watch out. Perimeter is found by adding; area is found by multiplying.
Method. Call the width w. The length is 3w. Add up the four sides to find w, then work out the area.
- Write the perimeter in terms of w. Two widths and two lengths.w + 3w + w + 3w = 8w
8w = 48 - Solve. Divide both sides by 8.w = 48 ÷ 8 = 6 cm
- Find the length. Length = 3 × width.3 × 6 = 18 cm
- Find the area. Length × width.18 × 6 = 108 cm²
Check. Perimeter check: 2 × (18 + 6) = 48 cm. ✓
Watch out. Read the last line of the question: it asks for the area, not for the width.
Method. Find the perimeter of the square, use it to find the width of the rectangle, then compare the two areas.
- Perimeter of the square. Four equal sides.4 × 8 = 32 cm
- Width of the rectangle. Same perimeter, so 2 lengths + 2 widths = 32.2 × 10 = 20
32 − 20 = 12 for the two widths
12 ÷ 2 = 6 cm - Area of the square. side × side.8 × 8 = 64 cm²
- Area of the rectangle. length × width.10 × 6 = 60 cm²
- Compare. 64 is bigger than 60.The square has the bigger area (by 4 cm²).
Check. Perimeter of the rectangle: 2 × (10 + 6) = 32 cm, the same as the square. ✓
Watch out. Shapes with the same perimeter can have different areas.
Method. Work out the area and the perimeter of each rectangle, then compare.
- Area of A. length × width.8 × 3 = 24 cm²
- Area of B. length × width.6 × 4 = 24 cm²
- Perimeter of A. Add all four sides.8 + 3 + 8 + 3 = 22 cm
- Perimeter of B. Add all four sides.6 + 4 + 6 + 4 = 20 cm
- Write the conclusion. State what you have shown.Both areas are 24 cm² but the perimeters are 22 cm and 20 cm, which are different.
Check. The area of each rectangle: A has 3 rows of 8 squares; B has 4 rows of 6 squares, and both make 24. ✓
Watch out. Always finish a “show that” question with a sentence.
Method. List every pair of whole numbers that multiply to 36 (the factor pairs), then work out the perimeter of each rectangle.
- List the factor pairs of 36. The length and width multiply to give the area, 36.1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
- Work out each perimeter. Perimeter = 2 × (length + width).1 by 36: 2 × 37 = 74 cm
2 by 18: 2 × 20 = 40 cm
3 by 12: 2 × 15 = 30 cm
4 by 9: 2 × 13 = 26 cm
6 by 6: 2 × 12 = 24 cm - Pick the smallest. Compare the five perimeters.The smallest perimeter is 24 cm, from the 6 cm by 6 cm square.
Check. Every rectangle above has area 36: for example 4 × 9 = 36. ✓
Watch out. Rectangles that are closer to a square have a smaller perimeter for the same area.
Method. Write the perimeter as an expression in x, set it equal to 28 and solve. Then find the sides and multiply.
- Write the perimeter. Add the four sides.(x + 4) + x + (x + 4) + x = 4x + 8
- Make an equation. The perimeter is 28 cm.4x + 8 = 28
- Solve it. Subtract 8, then divide by 4.4x = 20
x = 5 - Find the sides. Width = x, length = x + 4.width 5 cm, length 9 cm
- Find the area. Length × width.9 × 5 = 45 cm²
Check. Perimeter: 2 × (9 + 5) = 28. ✓
Watch out. Four sides give 4x (two of each), not 2x.
Method. A square has four equal sides. Find the side from the perimeter, then solve for y.
- Find the side length. Perimeter = 4 × side, so side = perimeter ÷ 4.44 ÷ 4 = 11 cm
- Make an equation. The side is (2y − 1).2y − 1 = 11
- Solve. Add 1, then divide by 2.2y = 12
y = 6 - Find the area. side × side.11 × 11 = 121 cm²
Check. 2 × 6 − 1 = 11. ✓
Watch out. Divide the perimeter by 4 first. Squaring the side would give the area, which is a different question.