G14, G16, G17Geometry & MeasuresFoundation & HigherGrades 1–5

Area & Perimeter

Perimeter is the distance around the outside of a shape; area is the amount of flat space inside it. You need the area formulas for rectangles, triangles, parallelograms and trapezia, and the problem-solving to split awkward shapes into these simple pieces.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Every area formula grows from the rectangle. A parallelogram rearranges into a rectangle, so its area is base × perpendicular height. A triangle is half of a parallelogram. A trapezium is two triangles, or the average of its parallel sides times the height.

The word $\textbf{perpendicular}$ is doing real work: the height must meet the base at a right angle. The slanted side of a parallelogram is a distractor that appears on many exam diagrams.

Area of Compound ShapesCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Calculate perimeters of shapes, including finding missing sides
Use formulas for the areas of rectangles, triangles, parallelograms and trapezia
Find areas and perimeters of compound shapes
Convert between area units such as cm² and m²

Perimeter and rectangles

Perimeter: add all the side lengths, including any that are not labelled — work these out from the others first.

Area of a rectangle $=$ length $\times$ width. Area is measured in square units: mm², cm², m², km².

Example 1A rectangle has area 84 cm² and length 12 cm. Find its perimeter.

1Width

= 84 \div 12 = 7

cm.

2Perimeter

= 2(12 + 7) = 38

cm.

Answer

38

Your turnSame method, new numbers — try it before moving on.

A square has a perimeter of 36 cm. What is its area, in cm²?

Your answer

Show worked solution

Solution

1Side $= 36 \div 4 = 9$ cm.
2Area $= 9 \times 9 = 81$ cm².

Answer: $81$ cm²

Triangles, parallelograms and trapezia

Triangle: $A = \frac{1}{2}bh$ . Parallelogram: $A = bh$ . Trapezium: $A = \frac{1}{2}(a + b)h$ , where $a$ and $b$ are the parallel sides.

In every formula, $h$ is the perpendicular height — at right angles to the base.

The dashed height is always perpendicular to the base — never the slanted side.

Example 1A trapezium has parallel sides 7 cm and 13 cm, and perpendicular height 6 cm. Find its area.

\frac{1}{2}(7 + 13) \times 6

= 10 \times 6 = 60

Answer

60

cm²

Your turnSame method, new numbers — try it before moving on.

A parallelogram has base 9 cm, slant side 5 cm and perpendicular height 4 cm. Find its area, in cm².

Your answer

Show hint

The slant side is not needed for the area.

Show worked solution

Solution

1 $A = bh = 9 \times 4 = 36$ cm².

Answer: $36$ cm²

Example 2A triangle has area 45 cm² and base 12 cm. Find its perpendicular height.

45 = \frac{1}{2} \times 12 \times h = 6h

h = 45 \div 6 = 7.5

Answer

7.5

Your turnSame method, new numbers — try it before moving on.

A trapezium has area 72 cm², height 8 cm and one parallel side of 7 cm. How long is the other parallel side, in cm?

Your answer

Show worked solution

Solution

1 $72 = \frac{1}{2}(7 + b) \times 8 = 4(7 + b)$ .
2 $7 + b = 18$ , so $b = 11$ cm.

Answer: $11$ cm

Compound shapes

Split the shape into rectangles, triangles and trapezia, find each area and add. Sometimes it is easier to subtract: find a large rectangle and take away the missing piece.

For the perimeter of a compound shape, work out every missing side and trace the whole outline.

Split along the dashed line: rectangle A is 6 by 4.5 and rectangle B is 13 by 4.5.

Example 1Find the area and perimeter of the L-shape above (all lengths in cm).

1A:

6 \times 4.5 = 27

; B:

13 \times 4.5 = 58.5

. Area

= 85.5

cm².

2Missing sides: top-right step

13 - 6 = 7

across and

9 - 4.5 = 4.5

down.

3Perimeter

= 6 + 4.5 + 7 + 4.5 + 13 + 9 = 44

cm.

Answer

Area

85.5

cm²; perimeter

44

Your turnSame method, new numbers — try it before moving on.

A rectangular lawn is 15 m by 8 m. A square flower bed 3 m by 3 m is cut out of it. What is the area of grass left, in m²?

Your answer

Show worked solution

Solution

1Lawn: $15 \times 8 = 120$ m². Bed: $3 \times 3 = 9$ m².
2Grass: $120 - 9 = 111$ m².

Answer: $111$ m²

Area units

$1$ cm $= 10$ mm, but $1$ cm² $= 10 \times 10 = 100$ mm². Likewise $1$ m² $= 100 \times 100 = 10{,}000$ cm².

Square the length conversion factor to convert area units.

Example 1Convert 3.5 m² to cm².

1

m²

= 10{,}000

cm².

3.5 \times 10{,}000 = 35{,}000

Answer

35{,}000

cm²

Your turnSame method, new numbers — try it before moving on.

Convert 4,800 cm² to m².

Your answer

Show worked solution

Solution

1Divide by $10{,}000$ : $4800 \div 10{,}000 = 0.48$ m².

Answer: $0.48$ m²

Why 10,000 and not 100

Picture a 1 m square covered in 1 cm squares: 100 rows of 100 squares. Dividing 35,000 cm² by 100 (as for lengths) would give a wildly wrong 350 m² — ten thousand is the factor for area.

Think like an examiner

Write the formula and the substitution before the answer — method marks depend on it.
Include square units for areas and linear units for perimeters.
For compound shapes, label each part (A, B) so your working is clear.

Area formulas

A_{\triangle} = \tfrac{1}{2}bh

Triangle.

A = bh

Parallelogram (and rectangle).

A = \tfrac{1}{2}(a + b)h

Trapezium:

a

b

are the parallel sides.

Watch out for these

Using the slant side as the height of a parallelogram.

Use the perpendicular height.

Forgetting to halve for a triangle.

A = \frac{1}{2} \times b \times h

Missing unlabelled sides when finding a compound perimeter.

Work out every side first, then add around the outline.

2

m²

= 200

cm².

2 \times 10{,}000 = 20{,}000

cm².

Stretch yourself

A rectangle has perimeter 34 cm and area 60 cm². Find its length and width.

Hint — Length + width = 17. Which two numbers add to 17 and multiply to 60?

Reveal solution

Solution

l + w = 17

and

lw = 60

. Trying factor pairs of 60:

12 \times 5

has sum 17 ✓. (Algebraically,

w(17 - w) = 60

gives

w^2 - 17w + 60 = 0

, so

(w - 5)(w - 12) = 0

.) The rectangle is 12 cm by 5 cm.

\blacksquare

Questions students ask

Do I need to learn the trapezium formula?

Yes — it is not given on AQA papers. Remember it as "the average of the parallel sides, times the height".

Why is the triangle formula half base times height?

Any triangle fits exactly into half of a parallelogram (or rectangle) with the same base and height.

What if a shape has curved edges?

Shapes with circular parts need the circle formulas — see the Circles lesson.

Key takeaways

Perimeter adds all sides; area measures the space inside.
Triangle $\frac{1}{2}bh$ ; parallelogram $bh$ ; trapezium $\frac{1}{2}(a + b)h$ .
Heights are always perpendicular to the base.
Split compound shapes, or subtract a missing piece.
Square the length factor to convert area units.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

A triangle has base 14 cm and perpendicular height 9 cm. Work out its area.[2]
Show mark scheme
- M1 $\frac{1}{2} \times 14 \times 9$
- A1 $63$ cm²
Answer: $63$ cm²

Question 2Grade 43 marks

A garden is a trapezium with parallel sides 18 m and 12 m, and a perpendicular distance of 10 m between them. Turf costs £4.50 per m² and is sold in whole square metres.

Work out the cost of turf for the whole garden.[3]
Show mark scheme
- M1 $\frac{1}{2}(18 + 12) \times 10$
- A1 $150$ m²
- A1£675
Answer: £675

Question 3Grade 53 marks

A square and a rectangle have the same area. The square has side 12 cm. The rectangle has width 8 cm.

Which shape has the greater perimeter, and by how much?[3]
Show mark scheme
- M1Area $144$ cm², so rectangle length $= 144 \div 8 = 18$ cm
- M1Perimeters: square $48$ cm, rectangle $52$ cm
- A1The rectangle, by 4 cm

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Area of a rectangle 9 cm by 6 cm?
- 54 cm²
- 30 cm²
- 15 cm²
- 27 cm²
Show answer and explanation
Answer: 54 cm²
$9 \times 6$ .
Perimeter of a rectangle 9 cm by 6 cm?
- 30 cm
- 54 cm
- 15 cm
- 36 cm
Show answer and explanation
Answer: 30 cm
$9 + 6 + 9 + 6$ .
Area of a triangle with base 10 cm and height 7 cm?
- 35 cm²
- 70 cm²
- 17 cm²
- 140 cm²
Show answer and explanation
Answer: 35 cm²
$\frac{1}{2} \times 10 \times 7$ .
Parallelogram: base 8 cm, slant side 6 cm, height 5 cm. Area?
- 40 cm²
- 48 cm²
- 30 cm²
- 20 cm²
Show answer and explanation
Answer: 40 cm²
Base × perpendicular height.
Trapezium: parallel sides 5 cm and 9 cm, height 4 cm. Area?
- 28 cm²
- 56 cm²
- 180 cm²
- 18 cm²
Show answer and explanation
Answer: 28 cm²
$\frac{1}{2} \times 14 \times 4$ .
Convert 5 cm² to mm².
- 500 mm²
- 50 mm²
- 5000 mm²
- 0.5 mm²
Show answer and explanation
Answer: 500 mm²
$1$ cm² $= 100$ mm².
A triangle has area 24 cm² and height 8 cm. Its base is…
- 6 cm
- 3 cm
- 16 cm
- 192 cm
Show answer and explanation
Answer: 6 cm
$24 = 4b$ .
A square has area 50 cm². Its perimeter is closest to…
- 28.3 cm
- 200 cm
- 12.5 cm
- 7.1 cm
Show answer and explanation
Answer: 28.3 cm
Side $\sqrt{50} \approx 7.07$ ; $4 \times 7.07$ .

Next step

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