Perimeter is one of the first geometry concepts you learn, but it still appears on every GCSE Maths paper — and the questions can be surprisingly tricky, especially when composite shapes or missing lengths are involved. AQA, Edexcel and OCR all test perimeter at both Foundation and Higher tiers, often combining it with algebra or circle calculations. This guide covers everything from basic polygons to circles and sectors, walks through worked examples, and provides practice questions so you are fully prepared. For a wider view of key GCSE formulas, check our GCSE Maths formulas guide.
What Is Perimeter?
The perimeter is the total distance around the outside of a two-dimensional shape. For straight-sided shapes, you simply add up all the side lengths. For circles, the perimeter has a special name: the circumference.
Key Formulas
Rectangle: $$P = 2(l + w)$$
Square: $$P = 4s$$
Triangle: $$P = a + b + c$$
Circumference of a circle: $$C = \pi d = 2\pi r$$
Arc length (part of a circle): $$\text{Arc} = \frac{\theta}{360} \times \pi d$$
where θ is the angle at the centre.
For any other polygon, add up all the side lengths.
Step-by-Step Method
Straight-Sided Shapes
- Identify all the side lengths. If some are missing, use the information given to work them out.
- Add up every side length.
- Give your answer with the correct unit (cm, m, etc.).
Circles
- Identify whether you have the radius or the diameter.
- Use C = πd or C = 2πr.
- Calculate and round as instructed.
Composite Shapes
- Identify the outer boundary — which edges form the perimeter?
- Calculate each straight section and any curved sections separately.
- Add them all together.
- Be careful: internal edges (where shapes meet) are not part of the perimeter.
Finding Missing Sides
In many questions, not all sides are labelled. Use these strategies:
- Opposite sides of a rectangle are equal.
- In L-shapes, the missing lengths can be found by subtracting one given length from another.
- In algebraic perimeter questions, add all expressions and simplify or set equal to a given total.
Worked Example 1 — Foundation Level
An L-shaped garden has dimensions as shown: the overall length is 14 m, overall width is 9 m, and the cut-out rectangle is 5 m by 4 m. Find the perimeter.
Step 1: Work out any missing sides.
- The horizontal lengths are: 14 m (top), 5 m (step in), and 14 − 5 = 9 m (bottom section).
- The vertical lengths are: 9 m (left), 4 m (step down), and 9 − 4 = 5 m (right section).
Step 2: Add all outer sides: 14 + 5 + 9 + 4 + 5 + 9 = 46 m.
(Note: an L-shape has 6 outer edges, not 4.)
Worked Example 2 — Higher Level
A shape is made from a rectangle (10 cm by 6 cm) with a semicircle added to one of the shorter ends. Find the perimeter of the shape. Give your answer to 1 decimal place.
Step 1: Identify the outer edges.
- Two long sides of the rectangle: 10 cm each.
- One short side of the rectangle (the end without the semicircle): 6 cm.
- The semicircle replaces the other short end: diameter = 6 cm.
Step 2: Semicircle circumference = ½ × πd = ½ × π × 6 = 3π = 9.4248 cm.
Step 3: Total perimeter = 10 + 10 + 6 + 9.4248 = 35.4 cm (1 d.p.).
Note: we do not include the straight edge where the semicircle meets the rectangle — that edge is internal.
Arc Length Example
Find the perimeter of a sector with radius 8 cm and angle 90°.
Arc length = (90/360) × π × 16 = ¼ × 16π = 4π = 12.566 cm.
Perimeter = arc + two radii = 12.566 + 8 + 8 = 28.6 cm (1 d.p.).
Common Mistakes
- Forgetting a side. In composite shapes, students often miss one of the edges. Count the number of sides carefully.
- Including internal edges. If two shapes are joined, the edge where they meet is not part of the perimeter.
- Using πr² instead of πd for circumference. πr² gives area, not circumference. The circumference formula is C = πd or 2πr.
- Not adding the straight edges of a sector. A sector's perimeter includes the arc and both radii.
- Mixing up radius and diameter. If given the diameter, do not double it again.
Exam Tips
- Label all the sides on the diagram, including missing ones you have calculated. This keeps your working clear.
- For algebraic perimeters, set up the expression by adding all sides, then simplify. If the perimeter is given, set your expression equal to it and solve.
- State the formula for circumference before substituting — this earns a method mark.
- For semicircles in composite shapes, remember the perimeter includes the curved part only (the diameter edge is usually internal or already counted).
- Double-check your answer by estimating. The perimeter of a shape should feel proportional to its size.
Practice Questions
Question 1 (Foundation) A rectangle has a length of 12 cm and a width of 5 cm. Find the perimeter.
Question 2 (Foundation) Find the circumference of a circle with radius 7 cm. Give your answer to 1 decimal place.
Question 3 (Higher) A running track consists of a rectangle 100 m by 60 m with a semicircle at each short end. Find the total distance around the track.
Question 4 (Higher) The perimeter of a triangle is 42 cm. The sides are x cm, (x + 3) cm and (2x − 1) cm. Find the value of x.
Question 5 (Higher) A sector has radius 12 cm and arc length 15 cm. Find the perimeter of the sector.
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Related Topics
- Area of 2D Shapes — area and perimeter are often tested together
- Circumference and Area of Circles — essential for curved perimeters
- Algebra — algebraic perimeter questions require forming and solving equations
- Composite Shapes — combining straight and curved edges
Summary
Perimeter is the total distance around the outside of a shape. For straight-sided shapes, add all side lengths. For circles, use C = πd. For sectors, add the arc length to both radii. In composite shapes, identify which edges are on the outside and be careful not to include internal edges. Label missing sides, show your formula, and always include units. Perimeter is straightforward once you are systematic, and it provides reliable marks across every GCSE paper.