Home › Learning Hub › IGCSE Maths › 5a Perimeter, area and circles
5a · E5.1–E5.3, E5.5

Units, perimeter, area and circles

Topic 5 Mensuration · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Convert between metric units of length, area, volume, capacity and mass, including cm² ↔ m² and m³ ↔ litres.
  • Find the perimeter and area of rectangles, triangles, parallelograms and trapezia (only the triangle formula is given).
  • Find the circumference and area of a circle, and give answers in terms of \( \pi \).
  • Find arc lengths and sector areas as fractions of the circle — for Core the angle is a factor of 360°; Extended includes any angle and major sectors EXTENDED.
  • Find perimeters and areas of compound shapes and parts of shapes.

📚The mathematics

Units

1 cm = 10 mm ⇒ 1 cm² = 100 mm², 1 cm³ = 1000 mm³
1 m = 100 cm ⇒ 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³
1 litre = 1000 cm³ = 1000 ml; 1 m³ = 1000 litres

Areas convert by the square of the length factor and volumes by the cube. A square metre is 100 cm by 100 cm, so it holds 10 000 square centimetres, not 100. Convert before calculating whenever the measurements are in mixed units.

Area formulas

rectangle: \( lw \)
triangle: \( \tfrac{1}{2}bh \) (given)
parallelogram: \( bh \)
trapezium: \( \tfrac{1}{2}(a + b)h \) (not given)

The height is always perpendicular to the base — never the slanted side.

A trapezium with parallel sides 7 cm and 11 cm and perpendicular height 5 cm marked as a dashed line with a right angle. The area is a half times (7 + 11) times 5 = 45 square centimetres.
\( \tfrac{1}{2}(7 + 11) \times 5 = 45 \) cm²: average of the parallel sides times the perpendicular height.

Circles, arcs and sectors

circumference \( = 2\pi r \) (given)
area \( = \pi r^{2} \) (given)
arc length \( = \dfrac{\theta}{360} \times 2\pi r \)
sector area \( = \dfrac{\theta}{360} \times \pi r^{2} \)

An arc or sector is just the fraction \( \tfrac{\theta}{360} \) of the whole circle. The perimeter of a sector is the arc plus two radii. “In terms of \( \pi \)” means leave \( \pi \) as a symbol: \( 18\pi \), not 56.5.

A sector of a circle with radius 9 cm and angle 80 degrees, shaded. The arc is labelled 4 pi cm and the sector area 18 pi square centimetres; the rest of the circle is drawn faintly.
Worked example (b): \( \tfrac{80}{360} \) of the circle. Arc \( 4\pi \) cm, area \( 18\pi \) cm².

Compound shapes

Split the shape into pieces you know, or subtract a piece from a larger one. For the perimeter, trace round the outside only — edges shared between pieces are not part of it.

A compound shape made from a 10 cm by 6 cm rectangle with a semicircle of diameter 6 cm attached to its right-hand end. The shared 6 cm edge is dashed because it is not part of the perimeter.
Practice question 7: rectangle plus semicircle. The dashed edge is inside the shape.

✏️Worked example

(a) (Non-calculator.) A trapezium has parallel sides 7 cm and 11 cm, 5 cm apart. Find its area. [2] (b) (Non-calculator.) EXTENDED A sector has radius 9 cm and angle \( 80^{\circ} \). Find its arc length and area in terms of \( \pi \), and its perimeter. [4] (c) (Non-calculator.) Change 3.5 m² into cm². [1]

(a) \( \tfrac{1}{2}(7 + 11) \times 5 = 45 \) cm².

(b) Arc \( = \tfrac{80}{360} \times 2\pi(9) = \tfrac{2}{9} \times 18\pi = 4\pi \) cm. Area \( = \tfrac{2}{9} \times 81\pi = 18\pi \) cm². Perimeter \( = 4\pi + 9 + 9 = 18 + 4\pi \) cm.

(c) \( 3.5 \times 10\,000 = 35\,000 \) cm².

Check it. \( \tfrac{80}{360} \) is a bit less than a quarter, and \( 18\pi \) is a bit less than a quarter of the whole circle’s \( 81\pi \) ✓.
Sector perimeter = arc only. The perimeter of a sector includes both radii. Leaving them out gives \( 4\pi \) instead of \( 18 + 4\pi \).

📝Practise

Questions in the style of the current papers. EXTENDED marks Extended-only content.

1. (Non-calculator.) Change (a) 4500 cm³ into litres, (b) 0.6 cm² into mm², (c) 2.4 m² into cm². [3]
(a) 4.5 litres. (b) 60 mm². (c) 24 000 cm².
2. (Non-calculator.) Find the area of a parallelogram with base 12 cm and perpendicular height 7 cm. [1]
\( 12 \times 7 = 84 \) cm².
3. (Calculator.) A circle has radius 6.5 cm. Find its circumference and its area. [2]
\( 2\pi(6.5) = 40.8 \) cm; \( \pi(6.5)^{2} = 133 \) cm² (3 s.f.).
4. (Non-calculator.) A sector has radius 6 cm and angle \( 120^{\circ} \). Find its area and arc length in terms of \( \pi \). [2]
\( \tfrac{1}{3} \) of the circle: area \( \tfrac{1}{3} \times 36\pi = 12\pi \) cm²; arc \( \tfrac{1}{3} \times 12\pi = 4\pi \) cm.
5. (Calculator.) Find the perimeter of a quarter circle of radius 8 cm. [2]
Arc \( \tfrac{1}{4} \times 16\pi = 4\pi \); perimeter \( = 4\pi + 16 = 28.6 \) cm.
6. (Calculator.) EXTENDED A sector of radius 8 cm has area 50 cm². Find the angle of the sector. [2]
\( \tfrac{\theta}{360} \times 64\pi = 50 \Rightarrow \theta = \dfrac{50 \times 360}{64\pi} = 89.5^{\circ} \).
7. (Calculator.) A shape is a 10 cm by 6 cm rectangle with a semicircle of diameter 6 cm on one short side (see the diagram). Find its area and perimeter. [4]
Area \( = 60 + \tfrac{1}{2}\pi(3)^{2} = 60 + 14.14 = 74.1 \) cm². Perimeter \( = 10 + 6 + 10 + \tfrac{1}{2}(2\pi \times 3) = 26 + 9.42 = 35.4 \) cm (the dashed side is not included).

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • Corbettmaths — arc length, sector area and compound shapes
  • Khan Academy — converting units of area and volume