Sheet № 212 · Foundation + Higher · AQA · Edexcel · OCR
Perimeter of Compound Shapes –
Perimeter of compound shapes is a frequently tested GCSE Maths topic at both Foundation and Higher tiers. A compound shape is made by combining two or more simple shapes such as rectangles, triangles, semicircles, or quarter circles. The perimeter is the total distance around the outside of the shape. This guide shows you how to find miss
§Key definitions
Question:
An L-shaped room has outer dimensions 10 m by 8 m. A 4 m by 3 m rectangle is cut from the top-right corner. Find the perimeter.
Answer:
The perimeter is 36 m.
Q1 (Foundation):
A T-shape has a horizontal bar 14 cm by 3 cm on top of a vertical bar 4 cm by 8 cm (centred). Find the perimeter.
Q2 (Foundation):
A rectangle 8 cm by 5 cm has a quarter circle of radius 5 cm cut from one corner. Find the perimeter to 1 d.p.
Q3 (Higher):
A shape is made of a rectangle 20 cm by 10 cm with semicircles on both short ends. Find the exact perimeter.
§Formulas to memorise
Circumference of a full circle = 2πr = πd
Semicircle arc length = πr
Quarter-circle arc length = πr / 2
Label all given dimensions — on the diagram.
Find missing lengths — in L-shapes and T-shapes, subtract known lengths from totals to find the gaps.
Identify every outer edge — trace your finger around the outside of the shape, listing each edge length.
Calculate curved edges — use πr for semicircles or πr/2 for quarter circles.
Add all outer edges — together to get the total perimeter.
Worked example
An L-shaped room has outer dimensions 10 m by 8 m. A 4 m by 3 m rectangle is cut from the top-right corner. Find the perimeter.
Working:
⚠ Common mistakes
- ✗Including internal edges. When two shapes are joined, the shared edge is inside the compound shape and must not be counted in the perimeter.
- ✗Forgetting the straight diameter edge of a semicircle. If a semicircle sits on top of a shape and its diameter is exposed (not shared), you must add the diameter as well as the arc.
- ✗Using diameter instead of radius in arc formulas. Semicircle arc = πr, not πd. Always halve the diameter first.
✦ Exam tips
- →Trace around the outside of the shape with your pen to make sure you count every edge exactly once.
- →For L-shapes and T-shapes, the perimeter is often equal to the perimeter of the enclosing rectangle — check this shortcut.
- →Always state the formula you are using for curved sections to earn method marks.
- →Give your answer in the units specified and round only at the final step.