Compound shapes that include curved parts — such as a rectangle with a semicircle on top, or a square with a circular hole — require you to break the shape into simpler components. Calculate the area or perimeter of each component separately, then combine the results, sometimes leaving answers in terms of π.
What formulae do you need?
| Shape | Area | Perimeter/Circumference |
|---|---|---|
| Circle (radius r) | πr² | 2πr |
| Semicircle | ½πr² | πr + 2r (arc + diameter) |
| Quarter-circle | ¼πr² | ½πr + 2r (arc + two radii) |
| Rectangle | length × width | 2(l + w) |
| Triangle | ½ × base × height | Sum of three sides |
| Sector (angle θ°) | (θ/360) × πr² | (θ/360) × 2πr + 2r |
Key reminder: The "perimeter" of a semicircle includes the straight diameter edge as well as the curved arc. Students frequently forget the straight edge — always list every boundary segment.
How do you find the area of a shape with a semicircle attached?
Worked example 1: A running track end-cap shape: a rectangle 20 m long and 14 m wide with a semicircle of diameter 14 m attached to each short end.
The total shape is a "stadium" — a rectangle with two semicircles (= one full circle).
- Area of rectangle: 20 × 14 = 280 m²
- Area of two semicircles (radius = 7 m): π × 7² = 49π m²
- Total area: (280 + 49π) m²
As a decimal: 280 + 153.94 ≈ 433.94 m² (3 s.f.)
As exact form: (280 + 49π) m²
How do you find the perimeter of a composite shape?
Worked example 2: A shape consists of a 10 cm × 6 cm rectangle with a semicircle of diameter 6 cm placed on one short end (6 cm side). Find the perimeter.
Identify every boundary segment:
- Two long sides of rectangle: 2 × 10 = 20 cm
- One short side of rectangle (opposite end from semicircle): 6 cm
- Arc of semicircle (radius = 3 cm): π × 3 = 3π cm (do NOT add the diameter — it is internal)
Total perimeter = 20 + 6 + 3π = (26 + 3π) cm
As a decimal: 26 + 9.42 ≈ 35.42 cm (4 s.f.)
Watch out: The 6 cm diameter at the join between the rectangle and semicircle is inside the shape — it is not a boundary edge. Only trace the outer boundary.
How do you find the area of a shape with a circle removed?
Worked example 3: A square of side 12 cm has a circle of radius 4 cm removed from its centre. Find the shaded area (the remaining region).
- Area of square: 12² = 144 cm²
- Area of circle: π × 4² = 16π cm²
- Shaded area: 144 − 16π = (144 − 16π) cm²
As a decimal: 144 − 50.27 ≈ 93.73 cm² (4 s.f.)
How do you find the area of a quarter-circle cutout?
Worked example 4: A 10 cm × 10 cm square has a quarter-circle of radius 10 cm removed from one corner. Find the remaining area.
- Area of square: 10² = 100 cm²
- Area of quarter-circle: ¼ × π × 10² = 25π cm²
- Remaining area: (100 − 25π) cm²
As a decimal: 100 − 78.54 ≈ 21.46 cm² (4 s.f.)
What does "leave your answer in terms of π" mean?
This instruction means do not substitute a decimal approximation for π. Write the answer in the form (a + bπ) or (a − bπ), where a and b are exact numbers.
Example: Instead of writing 26 + 9.424778…, write (26 + 3π).
Leaving answers in exact form avoids rounding errors and is always accepted as fully correct in GCSE mark schemes. When a decimal is needed (e.g. for a practical context), round to 3 significant figures unless told otherwise.
What mistakes should you avoid?
- Using diameter instead of radius in the area formula. πr² uses the radius — if diameter is given, halve it first.
- Forgetting to add the straight edge of a semicircle's perimeter. The perimeter includes the arc (πr) plus the flat diameter (2r).
- Adding areas when you should subtract (or vice versa). Always visualise whether the curved region is added to the shape or removed from it.
- Mixing exact and decimal values. If the question asks for exact form, the π must remain symbolic throughout — substituting 3.14 midway then writing "+ something π" is not exact.
Frequently asked questions
My answer has both a whole number part and a π part — can I simplify further?
Only if there is a common factor. For example, (20 + 4π) = 4(5 + π). However, 5 and π are not "like terms" — you cannot add them to get a single number. The expression (20 + 4π) is already in simplest form unless you need to factorise for a specific purpose.
Do I always need to find both area and perimeter?
Read the question carefully. "Find the area" and "find the perimeter" are different tasks; exam questions will specify which. Many students automatically calculate both and waste time. Some questions do ask for both — but only when explicitly requested.
How do I find the perimeter of a shape with a sector rather than a full semicircle?
For a sector of angle θ° and radius r: the arc length is (θ/360) × 2πr. Add the arc to the other boundary segments, being careful not to include the two radii if they are internal. For a quarter-circle (θ = 90°): arc length = (90/360) × 2πr = πr/2. The perimeter of a quarter-circle sector as an isolated shape includes the arc (πr/2) plus two radii (2r): total = πr/2 + 2r.
Can a compound shape include both a semicircle and a triangle?
Yes — for example, a "church window" shape: a rectangle with a semicircle on top and a triangle below. Split it into all three components, calculate each area, and add. For the perimeter, trace only the outer edge — the joins between components are interior boundaries and are excluded.
For GCSE geometry support from Professor Pi, visit AI Tutors for step-by-step help.