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Maths · Area and volume

Perimeter and area

The area formulae for rectangles, triangles, parallelograms and trapeziums, splitting compound shapes into simple parts, and working backwards from an area to a missing length.

  • 5 key terms
  • All boards
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Before We Start

Answer each one, then check.

  1. 1

    What is the perimeter of a 7 cm by 4 cm rectangle?

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    22 cm

  2. 2

    What is the area of a 7 cm by 4 cm rectangle?

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    28 cm²

  3. 3

    Work out \(\frac{1}{2} \times 9 \times 6\).

    Show answerHide answer

    27

  4. 4

    Solve \(4h = 24\).

    Show answerHide answer

    \(h = 6\)

Learning Objectives

  1. 1Find the perimeter of 2D shapes, including compound shapes.
  2. 2Use the area formulae for rectangles, triangles, parallelograms and trapeziums.
  3. 3Find the area of a compound shape.
  4. 4Work backwards from an area to a missing length.

Perimeter and Area

Perimeter is the distance round the edge; area is the space inside.

  • Perimeter

    Add every side. Units: cm, m.

  • Area

    Count squares or use a formula. Units: cm², m².

  • Perpendicular height

    Use the height at right angles to the base, NOT the slanted side.

  • Same units

    Change all the lengths into the same unit before you calculate.

Area of a Trapezium

A trapezium has parallel sides of 6 cm and 10 cm. The perpendicular distance between them is 5 cm. Work out its area.

Show the solutionHide the solution
  1. 1 Write the formula \(A = \frac{1}{2}(a + b)h\)
  2. 2 Substitute \(A = \frac{1}{2}(6 + 10) \times 5\)
  3. 3 Calculate \(\frac{1}{2} \times 16 \times 5 = 40\)

Answer40 cm²

A Compound Shape

An L-shape is made by cutting a 4 cm by 3 cm rectangle from one corner of a 10 cm by 8 cm rectangle. Work out its area and its perimeter.

Show the solutionHide the solution
  1. 1 Area: big rectangle minus the cut-out \(10 \times 8 - 4 \times 3 = 80 - 12 = 68\)
  2. 2 Perimeter: the sides are 10, 5, 4, 3, 6 and 8 \(10 + 5 + 4 + 3 + 6 + 8 = 36\)
  3. 3 Notice Cutting a corner out leaves the perimeter the same as the rectangle's: \(2(10 + 8) = 36\)

AnswerArea 68 cm², perimeter 36 cm

Working Backwards

A triangle has an area of 24 cm² and a base of 8 cm. Work out its perpendicular height.

Show the solutionHide the solution
  1. 1 Write the formula \(24 = \frac{1}{2} \times 8 \times h\)
  2. 2 Simplify \(24 = 4h\)
  3. 3 Solve \(h = 6\)

Answer6 cm

Common Mistakes

Wrong

  • Using the slanted side as the height of a parallelogram.
  • Forgetting the \(\frac{1}{2}\) for a triangle.
  • Mixing cm and m in one calculation.
  • Writing area in cm.

Right

  • Use the perpendicular height.
  • Triangle: \(\frac{1}{2} \times b \times h\).
  • Convert to one unit first.
  • Area is always in square units: cm², m².

Same Area, Different Shape

Draw as many different shapes as you can with an area of exactly 24 cm²: a rectangle, a triangle, a parallelogram and a trapezium. Label the dimensions of each. Which of your shapes has the smallest perimeter?

1. Choose a base.

2. Work out the height needed.

3. Compare the perimeters.

A good answer shows: For example: rectangle 6 by 4; triangle base 8, height 6; parallelogram base 6, height 4; trapezium with parallel sides 5 and 7 and height 4. Of the rectangles, the one closest to a square (about 4.9 by 4.9) has the smallest perimeter.

Can I...?

  1. 1Find the perimeter of a shape.
  2. 2Find the area of a rectangle and a triangle.
  3. 3Find the area of a parallelogram and a trapezium.
  4. 4Find missing sides of a compound shape.
  5. 5Find the area of a compound shape.
  6. 6Find a missing length from an area.

Summary & Exam Focus

  • Rectangle \(lw\); triangle \(\frac{1}{2}bh\); parallelogram \(bh\); trapezium \(\frac{1}{2}(a + b)h\).
  • Heights are always perpendicular.
  • Compound shapes: split, or subtract a missing part.
  • Area units are squared.

Exam focus

A wall is 4 m long and 2.5 m high. It has a door 0.8 m wide and 2 m high. The wall needs two coats of paint. One tin covers 5 m². How many tins are needed? (4 marks) (4 marks)

In a problem, write down what each number you calculate represents - "area of wall", "area to paint" - so the examiner can follow you and give process marks.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Perimeter
The total distance around the edge of a shape.
Area
The amount of space inside a 2D shape, in square units.
Perpendicular height
The height measured at right angles to the base.
Compound shape
A shape made from two or more simple shapes.
Trapezium
A quadrilateral with one pair of parallel sides.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    A trapezium has parallel sides of 7 cm and 11 cm and a perpendicular height of 6 cm. Work out the area of the trapezium.

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    Model answer

    \(\frac{1}{2}(7 + 11) \times 6 = 54\) cm²

    Mark scheme

    • \(\frac{1}{2}(7 + 11) \times 6\) — M1
    • 54 cm² — A1
  2. Question 2 Non-calculator 4 marks

    The diagram shows an L-shape. All the corners are right angles. Work out (a) the area and (b) the perimeter of the shape.

    An L-shape with a 10 cm bottom, 8 cm left side, 6 cm top and 5 cm right side, with the two inner sides unlabelled.
    Show answerHide answer

    Model answer

    The missing sides are \(10 - 6 = 4\) cm and \(8 - 5 = 3\) cm. (a) \(10 \times 8 - 4 \times 3 = 68\) cm² (b) \(10 + 5 + 4 + 3 + 6 + 8 = 36\) cm

    Mark scheme

    • Missing sides 4 and 3 found — P1
    • (a) A correct method for the area, e.g. \(80 - 12\) or \(6 \times 8 + 4 \times 5\) — P1
    • (a) 68 cm² — A1
    • (b) 36 cm — B1
  3. Question 3 Calculator 4 marks

    A wall is 4 m long and 2.5 m high. It has a door 0.8 m wide and 2 m high. The wall, but not the door, needs two coats of paint. One tin of paint covers 5 m² and costs £12. Work out the cost of the paint needed.

    Show answerHide answer

    Model answer

    Wall \(4 \times 2.5 = 10\) m². Door \(0.8 \times 2 = 1.6\) m². To paint \(10 - 1.6 = 8.4\) m², twice: 16.8 m². \(16.8 \div 5 = 3.36\), so 4 tins. \(4 \times 12 = £48\).

    Mark scheme

    • \(10 - 1.6 = 8.4\) — P1
    • \(8.4 \times 2 = 16.8\) — P1
    • 4 tins — P1
    • £48 — A1
  4. Question 4 Non-calculator 3 marks

    A triangle has a base of 12 cm and a perpendicular height of 9 cm. A rectangle has the same area as the triangle. The rectangle is 6 cm wide. Work out the perimeter of the rectangle.

    Show answerHide answer

    Model answer

    Triangle area \(\frac{1}{2} \times 12 \times 9 = 54\) cm². Rectangle length \(54 \div 6 = 9\) cm. Perimeter \(2(9 + 6) = 30\) cm.

    Mark scheme

    • 54 — P1
    • 9 cm — P1
    • 30 cm — A1

Quick check

  1. What is the area of a parallelogram with base 9 cm and perpendicular height 4 cm?

    1. A18 cm²
    2. B26 cm²
    3. C36 cm²
    4. D13 cm²
    Show answerHide answer

    C: 36 cm²

    Area = base × perpendicular height = \(9 \times 4\).

  2. What is the area of a triangle with base 10 cm and height 7 cm?

    1. A35 cm²
    2. B70 cm²
    3. C17 cm²
    4. D34 cm²
    Show answerHide answer

    A: 35 cm²

    \(\frac{1}{2} \times 10 \times 7 = 35\).

  3. A trapezium has area 60 cm², height 5 cm and one parallel side 10 cm. How long is the other parallel side?

    1. A12 cm
    2. B14 cm
    3. C2 cm
    4. D24 cm
    Show answerHide answer

    B: 14 cm

    \(60 = \frac{1}{2}(10 + b) \times 5\), so \(10 + b = 24\) and \(b = 14\).

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