Perimeter, area and volume | AQA GCSE Maths Notes

Visual model

Split a compound shape before finding area

area 1area 2split, find each area, then add

Gold-standard guide

20 mins

What you will learn

Calculate measurements for common shapes and solids.

Use a clear step-by-step method for perimeter, area and volume.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Area of trapezium = $\frac{1}{2}$ × (a + b) × h

Step 1

Identify the formula and values

Area = $\frac{1}{2}$ × (a + b) × h, where a = 7, b = 11, h = 8

Step 2

Substitute and calculate

Area = $\frac{1}{2}$ × (7 + 11) × 8 = $\frac{1}{2}$ × 18 × 8 = 72 cm²

Watch out

Students use the slant side instead of the perpendicular height

Area

$rectangle area = length \times width.$

Volume

$prism volume = cross-section area \times length.$

Worked example

Find the area of a trapezium with parallel sides 7 cm and 11 cm and a perpendicular height of 8 cm.

Identify the formula and values: Area = $\frac{1}{2}$ × (a + b) × h, where a = 7, b = 11, h = 8.

Substitute and calculate: Area = $\frac{1}{2}$ × (7 + 11) × 8 = $\frac{1}{2}$ × 18 × 8 = 72 cm².

Final answer

72 cm²

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Find the area of a trapezium with parallel sides 7 cm and 11 cm and a perpendicular height of 8 cm.

3 marks4 minsperimeter-area-and-volume-worked

Show solution

Worked solution

1.Identify the formula and values: Area = $\frac{1}{2}$ × (a + b) × h, where a = 7, b = 11, h = 8.
2.Substitute and calculate: Area = $\frac{1}{2}$ × (7 + 11) × 8 = $\frac{1}{2}$ × 18 × 8 = 72 cm².

Final answer

72 cm²

Mark points

M1: identify the formula and values
M1: substitute and calculate
A1: 72 cm²

Watch out

Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

Diagnosticrecall

Find the area of a triangle with base 14 cm and height 9 cm.

1 mark2 minsperimeter-area-and-volume-q1

Show solution

Worked solution

1.Spot the skill: Area of trapezium = $\frac{1}{2}$ × (a + b) × h.
2.Use the identify the formula and values stage first, then substitute and calculate.
3.Keep the final answer visible: 63 cm².

Final answer

63 cm²

Mark points

M1: use the correct area of trapezium = $\frac{1}{2}$ × (a + b) × h. area of triangle = $\frac{1}{2}$ × base × height.volume of prism = area of cross-section × length.always identify the perpendicular height — slant height is not the same.
A1: 63 cm²

Watch out

Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

Easyprocedure

Find the volume of a cuboid 5 cm × 4 cm × 3 cm.

2 marks3 minsperimeter-area-and-volume-q2

Show solution

Worked solution

1.Spot the skill: Area of trapezium = $\frac{1}{2}$ × (a + b) × h.
2.Use the substitute and calculate stage first, then identify the formula and values.
3.Keep the final answer visible: 60 cm³.

Final answer

60 cm³

Mark points

M1: use the correct area of trapezium = $\frac{1}{2}$ × (a + b) × h. area of triangle = $\frac{1}{2}$ × base × height.volume of prism = area of cross-section × length.always identify the perpendicular height — slant height is not the same.
A1: 60 cm³

Watch out

Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

Mediumreasoning

A prism has cross-section area 24 cm² and length 7 cm. Find its volume.

3 marks4 minsperimeter-area-and-volume-q3

Show solution

Worked solution

1.Spot the skill: Area of trapezium = $\frac{1}{2}$ × (a + b) × h.
2.Use the identify the formula and values stage first, then substitute and calculate.
3.Keep the final answer visible: 168 cm³.

Final answer

168 cm³

Mark points

M1: use the correct area of trapezium = $\frac{1}{2}$ × (a + b) × h. area of triangle = $\frac{1}{2}$ × base × height.volume of prism = area of cross-section × length.always identify the perpendicular height — slant height is not the same.
A1: 168 cm³

Watch out

Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

Hardproblem solving

Find the perimeter of a rectangle with length 13 cm and width 8 cm.

3 marks5 minsperimeter-area-and-volume-q4

Show solution

Worked solution

1.Spot the skill: Area of trapezium = $\frac{1}{2}$ × (a + b) × h.
2.Use the substitute and calculate stage first, then identify the formula and values.
3.Keep the final answer visible: 42 cm.

Final answer

42 cm

Mark points

M1: use the correct area of trapezium = $\frac{1}{2}$ × (a + b) × h. area of triangle = $\frac{1}{2}$ × base × height.volume of prism = area of cross-section × length.always identify the perpendicular height — slant height is not the same.
A1: 42 cm

Watch out

Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

Exam-stylemulti-step

A compound shape is a rectangle 10 × 6 cm with a right triangle of base 4 and height 6 removed from one end. Find the area.

4 marks6 minsperimeter-area-and-volume-q5

Show solution

Worked solution

1.Spot the skill: Area of trapezium = $\frac{1}{2}$ × (a + b) × h.
2.Use the identify the formula and values stage first, then substitute and calculate.
3.Keep the final answer visible: 48 cm².

Final answer

48 cm²

Mark points

M1: use the correct area of trapezium = $\frac{1}{2}$ × (a + b) × h. area of triangle = $\frac{1}{2}$ × base × height.volume of prism = area of cross-section × length.always identify the perpendicular height — slant height is not the same.
A1: 48 cm²

Watch out

Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

Grade 9 stretchproblem solving

A cylinder has radius 3 cm and height 8 cm. Find its volume in terms of $\pi$ .

4 marks7 minsmeasure-g9

Show solution

Worked solution

1.Use volume = $\pi r^{2}$ h.
2.Substitute the radius and height.

Final answer

$72pi c$ m³

Mark points

M1: use $\pi x$ 3² × 8
A1: $72pi c$ m³

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Perimeter, area and volume - 2 marksFind the area of a triangle with base 14 cm and height 9 cm.Mark answer

Answer

63 cm²

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer

Answer

Rectangle, square

4Circles - 3 marksFind the diameter of a circle with area

100pi c

m².Mark answer

Answer

20 cm

Mastery check

I can explain the method for perimeter, area and volume.
I can show clear working without skipping key steps.
I can avoid this mistake: Students use the slant side instead of the perpendicular height.The height must be the vertical distance between the two parallel sides.If not directly given, Pythagoras may be needed to find it.

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