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GCSE Maths · Geometry and measures

Area and perimeter GCSE Questions and Worked Answers

Perimeter measures the distance around a shape; area measures how much flat space it covers. Add outside edge lengths for perimeter, and count or calculate square units for area.

Foundation & Higher6 worked examples10 original questions
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Start with the meaning

What you need to know about area and perimeter

To cover a floor, you need to know how many square tiles fit inside it. To put trim around the floor, you need the distance around its edge. These are different measurements: area is the covering, and perimeter is the boundary length.

See the idea first

Count the inside and the outside separately

A 1 cm by 1 cm square covers one square centimetre, written cm². Four columns of three squares cover 12 cm². The boundary has four sides measuring 4, 3, 4 and 3 cm, giving 14 cm. Squared units measure area; ordinary length units measure perimeter.

4 cm3 cm
Each little square covers 1 cm². Count 4 × 3 = 12 squares inside, but 4 + 3 + 4 + 3 = 14 cm around the edge.
From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a rectangle's area and perimeter

What the problem asks: A rectangle measures 7 m by 4 m. Find both measures.

How to solve it: Area is seven rows of four square metres: 7 × 4 = 28 m². Perimeter counts all four edges: 7 + 4 + 7 + 4 = 22 m.

Find a triangle or trapezium area

What the problem asks: A triangle has base 10 cm and perpendicular height 6 cm. Find area.

How to solve it: Two copies fit a parallelogram of area 10 × 6, so one copy covers half: 30 cm². The height must be perpendicular to the base, not the sloping side.

Find a compound area

What the problem asks: An 8 cm by 6 cm rectangle has a 3 cm by 2 cm corner removed. Find remaining area.

How to solve it: Start with 48 cm² and subtract the missing 6 cm²: 42 cm². For perimeter, trace the new boundary instead of subtracting the cut-out's perimeter.

Measure a circle

What the problem asks: A circle has radius 3 cm. Find area and circumference.

How to solve it: Radius is centre to edge; diameter is twice it. Circumference = 2πr = 6π cm and area = πr² = 9π cm². π is the constant ratio of circumference to diameter.

A reliable routine

Partition a compound shape

For shapes made from familiar non-overlapping pieces, areas add because each square unit is counted once. You can instead subtract holes from a surrounding shape. Perimeter requires a separate boundary trace.

  1. Put all lengths in one unit and label any missing dimensions.
  2. For area, split into known shapes or subtract a cut-out.
  3. Use perpendicular heights: rectangle bh, triangle ½bh, parallelogram bh, trapezium ½(a + b)h for parallel sides a and b.
  4. For perimeter, count only the exposed boundary; finish with the appropriate units.

Check: Equal perimeters do not force equal areas. A diagram is not necessarily drawn to scale.

Fully worked

Area and perimeter GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Rectangle

2 marks
Question

Find the area and perimeter of a rectangle 9 cm long and 4 cm wide.

A=9×4=36 cm2A=9\times4=36\text{ cm}^2 P=9+4+9+4=26 cmP=9+4+9+4=26\text{ cm}

Tip: do not attach cm² to a perimeter.

Example 2

Triangle

2 marks
Question

A triangle has base 12 cm and perpendicular height 7 cm. Find its area.

The triangle covers half the corresponding parallelogram.

A=12×12×7A=\frac12\times12\times7 =42 cm2=42\text{ cm}^2

A sloping edge would not replace the perpendicular height.

Example 3

Trapezium

3 marks
Question

A trapezium's parallel sides measure 5 cm and 11 cm. Their perpendicular separation is 4 cm. Find its area.

Two matching trapezia form a parallelogram with base 5 + 11.

A=12(5+11)(4)A=\frac12(5+11)(4) =12(16)(4)=\frac12(16)(4) =32 cm2=32\text{ cm}^2
Example 4

L-shaped region

4 marks
Question

Find the area and perimeter of the L-shape.

8 cm6 cm3 cm2 cm
The missing top-right corner is 3 cm wide and 2 cm deep. A dashed line shows the original rectangle; it is not part of the remaining perimeter.

Area of outer rectangle minus cut-out:

A=8(6)−3(2)=48−6=42 cm2A=8(6)-3(2)=48-6=42\text{ cm}^2

The top edge is 8 − 3 = 5 cm; the right lower edge is 6 − 2 = 4 cm.

P=5+2+3+4+8+6=28 cmP=5+2+3+4+8+6=28\text{ cm}

The two new cut edges replace equal lengths removed from the outer boundary.

Example 5

Circle

3 marks
Question

A circle has diameter 10 cm. Find its circumference and area in terms of π.

Radius is half the diameter: r=5r=5 cm.

C=πd=10π cmC=\pi d=10\pi\text{ cm} A=πr2=π(5)2=25π cm2A=\pi r^2=\pi(5)^2=25\pi\text{ cm}^2

Tip: square the radius, not the diameter.

Example 6

A semicircle boundary

3 marks
Question

A semicircle has radius 4 cm. Find its area and perimeter in terms of π.

Area is half a circle:

A=12π(4)2=8π cm2A=\frac12\pi(4)^2=8\pi\text{ cm}^2

Perimeter includes the curved half-circumference and the straight diameter:

P=4π+8 cmP=4\pi+8\text{ cm}
10 original questions · total 24 marks

Area and perimeter GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 29 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

A square

2 marks

Find the area and perimeter of a square with side 6 cm.

Show worked answer
A=62=36 cm2A=6^2=36\text{ cm}^2 P=4(6)=24 cmP=4(6)=24\text{ cm}
2

Missing width

2 marks

A rectangle has area 63 cm² and length 9 cm. Find its width.

Show worked answer
w=63/9=7 cmw=63/9=7\text{ cm}

Area = length × width, so divide by the known length.

3

A parallelogram

2 marks

A parallelogram has base 13 cm, perpendicular height 5 cm and sloping side 6 cm. Find area.

Show worked answer

Use perpendicular height, not the side:

A=13×5=65 cm2A=13\times5=65\text{ cm}^2
4

Triangle height

2 marks

A triangle has area 36 cm² and base 9 cm. Find perpendicular height.

Show worked answer
36=12(9)h36=\frac12(9)h 72=9h72=9h h=8 cmh=8\text{ cm}
5

A trapezium

2 marks

Parallel sides of a trapezium are 8 m and 14 m, with perpendicular height 3 m. Find area.

Show worked answer
A=12(8+14)(3)=33 m2A=\frac12(8+14)(3)=33\text{ m}^2
6

Different units

3 marks

A rectangle is 2 m long and 75 cm wide. Find its area in m².

Show worked answer
75 cm=0.75 m75\text{ cm}=0.75\text{ m} A=2(0.75)=1.5 m2A=2(0.75)=1.5\text{ m}^2

Convert lengths before multiplying.

7

A rectangular hole

3 marks

A 10 m by 7 m lawn contains a 3 m by 2 m pond wholly inside it. Find the grass area.

Show worked answer
Alawn=10(7)=70A_{\rm lawn}=10(7)=70 Apond=3(2)=6A_{\rm pond}=3(2)=6 Agrass=70−6=64 m2A_{\rm grass}=70-6=64\text{ m}^2
8

Circle radius

3 marks

A circle has circumference 18π18\pi cm. Find its radius and area.

Show worked answer
2πr=18π2\pi r=18\pi r=9 cmr=9\text{ cm} A=π(9)2=81π cm2A=\pi(9)^2=81\pi\text{ cm}^2
9

Square-unit conversion

2 marks

Convert 2.4 m² to cm².

Show worked answer

A 1 m square is 100 cm by 100 cm, so 1 m² = 10,000 cm².

2.4×10000=24000 cm22.4\times10000=24000\text{ cm}^2
10

Same perimeter?

3 marks

A 5 cm by 5 cm square and an 8 cm by 2 cm rectangle have equal perimeters. Do they have equal areas?

Show worked answer

Both perimeters are 20 cm. The square's area is 25 cm², but the rectangle's is 16 cm². Equal boundary lengths do not determine equal areas.

Examiner-style feedback

Common area and perimeter mistakes

Area and perimeter swapped

Think covering versus edging and check square versus length units.

Sloping side used as height

The height in these area formulas meets the base at 90°.

Counting internal split lines

A line added to calculate area is not automatically part of the outside perimeter.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. Identify covering or boundary.
  2. Match units.
  3. Partition without overlap.
  4. Check whether the answer needs squared units.
Quick answers

Area and perimeter FAQ

Why does a triangle formula include one half?

Two matching triangles form a parallelogram with the same base and perpendicular height.

Must I use a calculator for π?

Not if the question asks for an answer in terms of π. Keep π exact until any requested rounding.

Build connected skills

What to revise next

Browse all GCSE Maths study guidesReview the worked examplesTry the questions again
Content standards

Curriculum and rights review

G16–G17 plane areas, perimeters, circles and compound shapes across tiers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references