GCSE — mathsFoundation8 min read

Area and Perimeter of 2D Shapes

Calculate area and perimeter of rectangles, triangles, circles, trapeziums and composite shapes for GCSE Maths with worked examples.

Tutor AI Team2026-02-24T03:53:42.207Z

Area and perimeter are two of the most practical topics in GCSE Maths. Perimeter is the total distance around the outside of a shape. Area is the amount of space inside a shape.

You need to know the formulae for common 2D shapes — rectangles, triangles, parallelograms, trapeziums and circles — and be able to apply them to solve problems, including composite (compound) shapes made from combinations of these.

Some formulae are given on the GCSE exam formula sheet, but knowing them from memory will save you time and help you feel more confident. This guide covers all the key shapes, their formulae, and how to handle composite shapes.

Core Concepts

Rectangle

$\text{Area} = l \times w$

$\text{Perimeter} = 2(l + w)$

where $l$ is the length and $w$ is the width.

Triangle

$\text{Area} = \frac{1}{2} \times b \times h$

where $b$ is the base and $h$ is the perpendicular height (the height measured at right angles to the base).

The perimeter is the sum of all three sides.

Parallelogram

$\text{Area} = b \times h$

where $b$ is the base and $h$ is the perpendicular height. Note: the slant height is not the perpendicular height.

Trapezium

$\text{Area} = \frac{1}{2}(a + b) \times h$

where $a$ and $b$ are the two parallel sides and $h$ is the perpendicular height between them.

This formula is given on the GCSE exam paper.

Circle

$\text{Area} = \pi r^2$

$\text{Circumference} = 2\pi r = \pi d$

where $r$ is the radius and $d$ is the diameter ( $d = 2r$ ).

The circumference is the perimeter of a circle.

Semicircle

$\text{Area of semicircle} = \frac{1}{2} \pi r^2$

$\text{Perimeter of semicircle} = \pi r + 2r = \pi r + d$

The perimeter includes the curved part and the diameter.

Composite Shapes

A composite shape is made from two or more simple shapes combined. To find the area:

Split the shape into simpler parts (rectangles, triangles, etc.)
Calculate the area of each part
Add (or subtract) as needed

For the perimeter of a composite shape, add up all the outer edges. Be careful not to include internal edges.

Units

Perimeter is measured in units of length: mm, cm, m, km.

Area is measured in square units: mm², cm², m², km².

Always check your units are consistent before calculating.

Strategy Tips

Tip 1: Memorise the Core Formulae

The rectangle, triangle and circle formulae are essential. The trapezium formula is given on the exam paper, but it's faster if you know it.

Tip 2: Always Identify the Perpendicular Height

For triangles, parallelograms and trapeziums, the height must be perpendicular to the base. If you use the slant height instead, your answer will be wrong.

Tip 3: Use $\pi = 3.14159...$ or the $\pi$ Button

Unless told otherwise, use the $\pi$ button on your calculator for accuracy. If the question says "give your answer in terms of $\pi$ ", leave $\pi$ in your answer (e.g., $25\pi$ cm²).

Tip 4: Draw Lines on Composite Shapes

When working with compound shapes, draw dashed lines to show how you have split the shape. This helps you see the dimensions and avoids errors.

Tip 5: Don't Confuse Area and Perimeter

Read the question carefully. "Find the area" and "find the perimeter" require different calculations. It's a surprisingly common exam mistake to calculate the wrong one.

Worked Example: Example 1

Problem

Find the area and perimeter of a rectangle with length 12 cm and width 5 cm.

Solution

$\text{Area} = 12 \times 5 = 60 \text{ cm}^2$

$\text{Perimeter} = 2(12 + 5) = 2 \times 17 = 34 \text{ cm}$

Worked Example: Example 2

Problem

Find the area of a triangle with base 9 cm and perpendicular height 6 cm.

Solution

$\text{Area} = \frac{1}{2} \times 9 \times 6 = 27 \text{ cm}^2$

Worked Example: Example 3

Problem

A trapezium has parallel sides of 8 cm and 14 cm, and a perpendicular height of 5 cm. Find its area.

Solution

$\text{Area} = \frac{1}{2}(8 + 14) \times 5 = \frac{1}{2} \times 22 \times 5 = 55 \text{ cm}^2$

Worked Example: Example 4

Problem

Find the area and circumference of a circle with radius 7 cm. Give your answers to 1 decimal place.

Solution

$\text{Area} = \pi \times 7^2 = 49\pi = 153.9 \text{ cm}^2 \text{ (1 d.p.)}$

$\text{Circumference} = 2 \times \pi \times 7 = 14\pi = 44.0 \text{ cm (1 d.p.)}$

Worked Example: Example 5

Problem

An L-shaped room is made from two rectangles. The overall dimensions are 10 m by 8 m, with a rectangular section of 4 m by 3 m cut from one corner. Find the area of the room.

Solution

Area of full rectangle: $10 \times 8 = 80 \text{ m}^2$

Area of cut-out: $4 \times 3 = 12 \text{ m}^2$

$\text{Area of L-shape} = 80 - 12 = 68 \text{ m}^2$

Worked Example: Example 6

Problem

A shape consists of a rectangle with a semicircle on one of its shorter ends. The rectangle is 10 cm long and 6 cm wide. Find the total area and perimeter of the shape. Give answers to 1 decimal place.

Solution

The semicircle has diameter 6 cm, so radius $r = 3$ cm.

Area:

$\text{Rectangle} = 10 \times 6 = 60 \text{ cm}^2$

$\text{Semicircle} = \frac{1}{2} \pi \times 3^2 = \frac{9\pi}{2} = 14.1 \text{ cm}^2$

$\text{Total area} = 60 + 14.1 = 74.1 \text{ cm}^2$

Perimeter:

The perimeter consists of: two long sides (10 cm each), one short side (6 cm), and the semicircular arc.

$\text{Semicircular arc} = \frac{1}{2} \times 2\pi \times 3 = 3\pi = 9.4 \text{ cm}$

$\text{Total perimeter} = 10 + 10 + 6 + 9.4 = 35.4 \text{ cm}$

Practice Problems

Problem 1

Find the area of a parallelogram with base 11 cm and perpendicular height 8 cm.

Problem 2

A circle has a diameter of 18 cm. Find its area and circumference, giving answers to 1 decimal place.

Problem 3

A trapezium has parallel sides of 6 cm and 10 cm and a height of 7 cm. Find its area.

Problem 4

A rectangular garden measures 15 m by 9 m. A circular pond with radius 2 m is in the centre. Find the area of the garden not covered by the pond, correct to 1 decimal place.

Problem 5

Find the perimeter of a semicircle with diameter 12 cm. Give your answer to 1 decimal place.

Problem 6

A compound shape is made of a rectangle (8 cm × 5 cm) with an equilateral triangle (side 5 cm) attached to one of the shorter sides. Find the total area of the shape. (Hint: the height of an equilateral triangle with side $s$ is $\frac{s\sqrt{3}}{2}$ .)

Want to check your answers and get step-by-step solutions?

Common Mistakes

Using diameter instead of radius (or vice versa) for circles. $\text{Area} = \pi r^2$ uses the radius, not the diameter. If given the diameter, halve it first.
Using slant height instead of perpendicular height. For triangles, parallelograms and trapeziums, the height must be at right angles to the base.
Forgetting the straight edge in semicircle perimeters. The perimeter of a semicircle is not just the curved part; you must add the diameter.
Not squaring the units for area. If the sides are in cm, the area is in cm², not cm.
Adding internal edges to the perimeter. When finding the perimeter of a composite shape, only include the outer boundary.

Frequently Asked Questions

Which formulae are given on the exam paper?

The GCSE formula sheet typically gives the area of a trapezium and the volume/surface area of some 3D shapes. The area formulae for rectangles, triangles and circles are generally expected to be known. Always check the front of your exam paper.

What does "give your answer in terms of $\pi$" mean?

It means leave $\pi$ as a symbol in your answer rather than converting to a decimal. For example, write $36\pi$ cm² instead of $113.1$ cm².

How do I find the area of an irregular shape?

Split it into shapes you know (rectangles, triangles, circles, etc.), calculate each area, then add or subtract as needed.

What is the difference between perimeter and circumference?

Circumference is the specific name for the perimeter of a circle. They both mean the total distance around the outside.

How do I convert between cm² and m²?

There are 100 cm in 1 m, so $1 \text{ m}^2 = 10\,000 \text{ cm}^2$ . To convert from cm² to m², divide by 10,000.

Key Takeaways

✓
Perimeter = distance around. Add up all the outer edges of the shape.
✓
Area = space inside. Use the correct formula for each shape.
✓
Perpendicular height is essential. For triangles, parallelograms and trapeziums, always use the height at right angles to the base.
✓
Circle formulae use radius. $A = \pi r^2$ and $C = 2\pi r$ . If given the diameter, halve it first.
✓
Composite shapes: split and combine. Break compound shapes into simpler parts, find each area, then add (or subtract) them together.
✓
Watch your units. Perimeter is in cm/m; area is in cm²/m². Always check units are consistent.

Tags:

#gcse#maths#geometry#area#perimeter

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Core Concepts

Rectangle

Triangle

Parallelogram

Trapezium

Circle

Semicircle

Composite Shapes

Units

Strategy Tips

Tip 1: Memorise the Core Formulae

Tip 2: Always Identify the Perpendicular Height

Tip 3: Use π=3.14159...\pi = 3.14159...π=3.14159... or the π\piπ Button

Tip 4: Draw Lines on Composite Shapes

Tip 5: Don't Confuse Area and Perimeter

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: Example 4

Worked Example: Example 5

Worked Example: Example 6

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Expanding and Factorising

Fractions, Decimals and Percentages

Pythagoras' Theorem

Ready to Ace Your GCSE maths?

Tip 3: Use $\pi = 3.14159...$ or the $\pi$ Button