Contents · 0% complete

Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn8300Papers 1–3 · Higher tierGeometry and measuresPerimeter, area and units

Geometry and measures · Subtopic 6 of 17

Perimeter, Area and Units

Perimeter is the distance around a shape; area is the space inside it. Five area formulas cover almost every straight-sided shape, and compound shapes are just these pieces added or subtracted. When converting area units, square the length conversion.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find the perimeter and area of rectangles, triangles, parallelograms and trapezia.
Split compound shapes into simple pieces.
Convert between units of area ( $\text{cm}^2$ , $\text{m}^2$ ) and volume ( $\text{cm}^3$ , $\text{m}^3$ , litres).
Work backwards from an area to a missing length.

Pause and think

A $1$ m by $1$ m square is $100$ cm by $100$ cm. How many $1$ cm squares fit inside it?

Section 2 of 14

Core idea

Start hereEverything is a rectangle

The only area most people really "know" is length × width. Every other formula comes from cutting and rearranging shapes into rectangles: a triangle is half a rectangle, a parallelogram is a slanted one.

The big idea

Triangle $\frac12bh$ ; parallelogram $bh$ ; trapezium $\frac12(a+b)h$ . Always the perpendicular height.

1. Formulas

A triangle is half of a rectangle

The triangle fills exactly half the rectangle around it. A trapezium's area uses the average of its parallel sides times the height.

From rectangles

$\text{parallelogram}=bh$
Cut a triangle off one end, slide it to the other: a rectangle.
$\text{triangle}=\tfrac12bh$
Two copies form a parallelogram.
$\text{trapezium}=\tfrac12(a+b)h$
Two copies (one flipped) form a parallelogram of base $a+b$ .

Worked exampleTriangle and trapezium

Triangle $b=10$ , $h=6$ ; trapezium $a=5$ , $b=9$ , $h=4$ .

Goal: Areas.
Strategy: Formulas.

$\tfrac12(10)(6)=30$
Triangle.
$\tfrac12(5+9)(4)=28$
Trapezium.

Answer

30

cm²,

28

cm².

✓ CheckTrapezium: average width

7

× height

4

✓.

Worked exampleL-shape

$10\times8$ rectangle with a $4\times3$ corner removed.

Goal: Area and perimeter.
Strategy: Subtract.

$80-12=68$
Area.
$2(10+8)=36$
Perimeter: the notch's two edges equal the missing ones.

Answer

68

cm²;

36

cm.

✓ CheckSplit instead:

10\times5+6\times3=68

✓.

Common misconceptionSlant height For a parallelogram use the perpendicular height, not the slanted side.

Pause and think

Triangle area 24, base 8. Height?

Summary

Perpendicular height.
Split or subtract compound shapes.

Perpendicular height: Height at right angles to the base.

HistoryEgyptian rope-stretchers. When the Nile flooded each year it washed away field boundaries. Egyptian surveyors, called "rope-stretchers", re-measured the land with knotted ropes so farmers could be taxed fairly on their area. The word geometry means "earth measurement".

Fun factSame perimeter, different area. A

1\times9

rectangle and a

5\times5

square both have perimeter

20

, but areas

9

and

25

. For a fixed perimeter, the square always gives the biggest rectangle; this is why farmers fencing a field prefer squarish shapes.

Section 3 of 14

Key formulae

Rectangle

A=lw

Triangle

A=\tfrac12bh

Parallelogram

A=bh

Trapezium

A=\tfrac12(a+b)h

Unit conversions

1\text{ m}^2=10^4\text{ cm}^2,\quad1\text{ m}^3=10^6\text{ cm}^3,\quad1\text{ litre}=1000\text{ cm}^3

TrickSquare it, cube it.

1

=100

cm, so for area use

100^2

and for volume use

100^3

. The same works for any unit:

1

=10

mm, so

1

^2=100

^2

and

1

^3=1000

^3

Section 4 of 14

Worked examples

Example 1 · Compound shape0 / 2 steps

A rectangle

10

cm by

8

cm has a

4

cm by

3

cm rectangle cut from one corner. Find the area of the L-shape.

Example 2 · Converting units0 / 1 steps

Convert

2500

^2

to m

^2

Section 5 of 14

Explore the graph

Rectangles with perimeter 20

Loading the interactive graph…

Try this

Every rectangle here has perimeter $20$ : width $w$ , length $10-w$ . The point shows its area.
Which width gives the largest area? What shape is that rectangle?
Why is the graph symmetrical?

For a fixed perimeter, the area is greatest for a square.

In the real worldBuying paint and carpet. Decorators measure walls and floors in m

^2

because paint tins and carpet are sold by area: "covers

12

^2

per litre". Getting the unit conversion wrong by a factor of

10\,000

would be an expensive mistake.

Section 6 of 14

Common mistakes

Incorrect

1

^2=100

^2

Correct

10\,000

^2

Square the length conversion.

IncorrectUsing the slanted side as the height

CorrectUse the perpendicular height

Height is measured at right angles to the base.

IncorrectTriangle area

=bh

Correct

\tfrac12bh

A triangle is half a parallelogram.

IncorrectPerimeter in cm

^2

Correctcm

Perimeter is a length.

$1\text{ m}^2=100\text{ cm}^2$

1

=100

and a square has two sides

$10\,000\text{ cm}^2$ . The square was square all along.

Section 7 of 14

Exam tip

Exam tipShow the split. For compound shapes, write the area of each piece separately with its calculation before adding. If you make one slip, you still earn the method marks for the rest.

Section 8 of 14

Guided practice

SA-4024-ME-G012 marksL1 FoundationNon-calculatorPerimeter & area

A trapezium has parallel sides of

7

cm and

11

cm, and a perpendicular height of

5

cm. Find its area.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-4024-ME1-0011 markL1 FoundationNon-calculatorPerimeter & area

Find the perimeter of a rectangle

8

cm long and

5

cm wide.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0021 markL1 FoundationNon-calculatorPerimeter & area

A triangle has base

10

cm and perpendicular height

7

cm. Find its area.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0032 marksL2 StandardNon-calculatorPerimeter & area

A parallelogram has base

9

cm, perpendicular height

4

cm and slanted sides of

5

cm. Find its area.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0042 marksL2 StandardNon-calculatorPerimeter & area

A rectangle has area

48

^2

and width

6

cm. Find its perimeter.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME-0011 markL1 FoundationNon-calculatorUnits

Convert

3.5

^2

to cm

^2

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0051 markL1 FoundationNon-calculatorUnits

Convert

2500

^2

to m

^2

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0061 markL1 FoundationNon-calculatorUnits

Convert

4.2

litres to cm

^3

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0071 markL2 StandardNon-calculatorUnits

Convert

0.6

^3

to cm

^3

Salam Academy Original Practice · Proposed mark scheme

SA-4024-ME1-0082 marksL2 StandardNon-calculatorPerimeter & area

A rectangle

10

cm by

8

cm has a

4

cm by

3

cm rectangle cut from one corner. Find the area of the remaining L-shape.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-4024-ME-0133 marksL2 StandardCalculatorPerimeter & area

A shape is a rectangle

10

cm by

6

cm with a semicircle on one of its

6

cm sides, on the outside. Find the area of the shape, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Rectangle $lw$ , triangle $\tfrac12bh$ , parallelogram $bh$ , trapezium $\tfrac12(a+b)h$ .
Use perpendicular heights.
Compound shapes: add or subtract simple pieces.
Area conversions use $100^2$ ; volume conversions use $100^3$ ; $1$ litre $=1000$ cm $^3$ .

Section 12 of 14

Quick quiz

Q1 · SA-4024-ME-Q011 markL1 FoundationCalculatorUnits

1

^3

is equal to

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-4024-ME1-0091 markL1 FoundationCalculatorPerimeter & area

A square has area

49

^2

. Its perimeter is

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: GCSE Mathematics 8300
Where: Geometry and measures
Expected: The same shapes and conversions, including compound units.
Calculator: Paper 1 non-calculator; Papers 2 and 3 calculator.
Typical style: Problem-solving questions often combine topics.
Open in: 8300 · Papers 1–3 8300 · Papers 1–3

Section 14 of 14

Past paper questions

4 questions (14 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

4024/22 · May/June 2025Question 51 mark
Convert an area from m² to cm².
Question paper, page 4 Mark scheme
4024/12 · May/June 2024Question 42 marks
Converting metric units.
Question paper, page 4 Mark scheme
4024/12 · May/June 2024Question 53 marks
Perimeter of a parallelogram and area of a trapezium.
Question paper, page 5 Mark scheme
4024/22 · May/June 2024Question 18 marks
Unit costs, reading a graph, unit conversion and ratio.
Question paper, page 2 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).