IGCSE Maths · Mensuration · Core & Extended

Units, Area, Perimeter and Circles

Mensuration is the maths of measuring — lengths, areas, and (in the next note) volumes of real shapes. This note covers unit conversions, the area/perimeter formulas for flat shapes, and everything to do with circles. Almost all of it is shared between tiers; the one real split is that Core restricts sector questions to angles that divide evenly into 360°, while Extended allows any angle and explicitly covers major sectors.

1. Units of Measure and Conversions

QuantityUnitsConversion
Lengthmm, cm, m, km10mm=1cm, 100cm=1m, 1000m=1km
Areamm², cm², m², km²100mm²=1cm², 10,000cm²=1m²
Volumemm³, cm³, m³1,000mm³=1cm³, 1,000,000cm³=1m³
Capacityml, l1000ml=1l, and 1cm³=1ml
Memory tip — “square it for area, cube it for volume” Whatever the length conversion factor is, area uses it squared, and volume uses it cubed — because area is length×length, and volume is length×length×length. So if 1 m = 100 cm, then:
1 m² = 100² = 10,000 cm²    1 m³ = 100³ = 1,000,000 cm³
This one rule generates every area and volume conversion in the table above — no separate memorising needed.
Worked examples Convert 3.5 km to metres.
3.5 × 1000 = 3500 m

Convert 2.5 m² to cm².
2.5 × 10,000 = 25,000 cm² (squaring the 100 cm-per-m conversion)

Convert 50,000 mm² to cm².
50,000 ÷ 100 = 500 cm²

Convert 3.2 m³ to litres.
3.2 × 1,000,000 cm³ = 3,200,000 cm³, and 1 cm³ = 1 ml, so = 3,200,000 ml = 3200 litres
Try it — Unit Conversions
  1. Convert 4200 m to km.
  2. Convert 6 m² to cm².
  3. Convert 2 m³ to cm³.
  4. Convert 4500 ml to litres.
Show answers
  1. 4.2 km
  2. 60,000 cm²
  3. 2,000,000 cm³
  4. 4.5 litres

2. Area and Perimeter of Rectangles, Triangles, Parallelograms and Trapeziums

Rectangle: A = l × w,   P = 2(l + w)
not given
Triangle: A = ½bh
given on formula sheet
Parallelogram: A = bh
not given
Trapezium: A = ½(a + b)h
not given
Memory tips
  • Triangle is the one exception — of these four, only the triangle formula (A = ½bh) is printed on the exam formula sheet. Rectangle, parallelogram, and trapezium you need to know from memory.
  • Triangle = half a rectangle. Picture a triangle sitting inside a rectangle with the same base and height — it always fills exactly half of it. That’s where the ½ comes from.
  • Parallelogram = a “pushed over” rectangle. Same formula shape as a rectangle (base × height), just remember it’s the perpendicular height, not the slanted side.
  • Trapezium: “add the parallel sides, average them, then stretch.” Add the two parallel sides, halve that (find the average), then multiply by the height — you’re essentially using the “average width” of the shape.
Trapezium example 8 cm 14 cm 5 cm Find the area of a trapezium with parallel sides 8 cm and 14 cm, and height 5 cm.
A = ½(8 + 14) × 5 = ½ × 22 × 5 = 55 cm²
More worked examples Rectangle 8 cm × 5 cm: Area = 40 cm², Perimeter = 2(8+5) = 26 cm
Triangle base 10 cm, height 6 cm: Area = ½ × 10 × 6 = 30 cm²
Parallelogram base 12 cm, perpendicular height 7 cm: Area = 12 × 7 = 84 cm²
Try it — Area and Perimeter
  1. Find the area and perimeter of a rectangle 9 cm by 6 cm.
  2. Find the area of a triangle with base 14 cm and height 9 cm.
  3. Find the area of a parallelogram with base 15 cm and height 8 cm.
  4. Find the area of a trapezium with parallel sides 6 cm and 10 cm, height 4 cm.
Show answers
  1. Area = 54 cm², Perimeter = 30 cm
  2. 63 cm²
  3. 120 cm²
  4. 32 cm²

3. Circumference and Area of a Circle

Circumference: C = 2πr  (or πd)
Area: A = πr²

Use the π button on your calculator for full accuracy, or 3.142 if told to.

Memory tips
  • Both circle formulas are given on the exam formula sheet — you don’t need to memorise C=2πr or A=πr², but you do need to recognise which one to use.
  • “Pi are squared” — the classic joke helps the area formula stick: circle area uses r squared (πr²), not just r.
  • Circumference has no square — it’s a length (a distance around), so it only needs one factor of r: C = 2πr.
  • If you mix up the two, check the units required in the question — a length answer (cm) needs circumference; an area answer (cm²) needs the squared formula.
Worked example A circle has radius 7 cm. Find its circumference and area.
Circumference = 2π(7) = 44.0 cm (3 s.f.)
Area = π(7)² = 153.9 cm² (3 s.f.)
Try it — Circles
  1. Find the circumference of a circle with radius 10 cm.
  2. Find the area of a circle with radius 5 cm.
  3. A circle has diameter 18 cm. Find its circumference.
  4. A circle has area 78.5 cm². Find its radius (use π ≈ 3.14).
Show answers
  1. 2π(10) = 62.8 cm
  2. π(5)² = 78.5 cm²
  3. π(18) = 56.5 cm
  4. r² = 78.5÷3.14 = 25, so r = 5 cm

4. Arc Length and Sector Area

A sector is a “pizza slice” of a circle — its arc length and area are simply fractions of the full circumference and area, based on the sector’s angle out of 360°.

Arc length = (θ/360) × 2πr
Sector area = (θ/360) × πr²
Memory tip Both of these are also given on the formula sheet — but only as the plain circumference/area formulas, not with the (θ/360) fraction attached. You have to remember to bolt that fraction on yourself. Think of it as: “what fraction of the full pizza is my slice?” — work that fraction out first (angle ÷ 360), then apply it to whichever full-circle formula matches what you’re finding (length → circumference formula; area → area formula).
Worked example 90° 10 cm Find the arc length and sector area for a sector of radius 10 cm and angle 90°.
Arc length = (90/360) × 2π(10) = ¼ × 62.83 = 15.7 cm
Sector area = (90/360) × π(10)² = ¼ × 314.16 = 78.5 cm²
Worked example — an angle that isn’t a factor of 360° Extended Find the arc length and sector area for a sector of radius 8 cm and angle 137°.
Arc length = (137/360) × 2π(8) = 19.1 cm (3 s.f.)
Sector area = (137/360) × π(8)² = 76.5 cm² (3 s.f.)
Worked example — a major sector Extended A sector has a minor angle of 80°. Find the area of the corresponding major sector, radius 6 cm.
Major sector angle = 360 − 80 = 280°
Major sector area = (280/360) × π(6)² = 88.0 cm² (3 s.f.)
Try it — Arc Length and Sector Area
  1. Find the arc length of a sector with radius 6 cm and angle 60°.
  2. Find the sector area for radius 12 cm and angle 45°.
  3. Find the arc length for radius 9 cm and angle 200° (an angle over 180°). Ext.
  4. A sector has minor angle 110° and radius 5 cm. Find the major sector’s area. Ext.
Show answers
  1. (60/360)×2π(6) = 6.28 cm
  2. (45/360)×π(12)² = 56.5 cm²
  3. (200/360)×2π(9) = 31.4 cm
  4. Major angle=250°. (250/360)×π(5)² = 54.5 cm²
Common mistakes to watch for
  • Using diameter instead of radius (or vice versa) in the area formula — area always uses radius, never diameter directly.
  • Forgetting to square the conversion factor when converting between area units (e.g. cm² to m² needs ÷10,000, not ÷100).
  • Using the trapezium’s slanted side length instead of the perpendicular height in the area formula.
  • Forgetting the sector angle fraction (θ/360) entirely, and just using the full circle formulas.
  • Mixing up a minor and major sector — check whether the angle given is the small one or whether you need to subtract from 360° first.

5. Quick-Fire Challenge Round

True or False?
  1. 1 m² = 100 cm².
  2. The area of a parallelogram uses the perpendicular height, not a slanted side.
  3. Arc length is a fraction of the circle’s area.
  4. A major sector’s angle plus its corresponding minor sector’s angle always equals 360°.
Show answers
  1. False — 1 m² = 10,000 cm², since both length dimensions are converted (100×100).
  2. True.
  3. False — arc length is a fraction of the circumference; sector area is a fraction of the area.
  4. True. Ext.
Spot the Error
  1. “Convert 3 m² to cm²: 3 × 100 = 300 cm².”
  2. “Area of a trapezium with parallel sides 6 and 10, height 4: A = 6 × 10 × 4.”
  3. “Sector area, radius 6, angle 90°: A = π × 6² = 113.1 cm².”
Show answers
  1. Area conversion needs ×10,000 (100²), not ×100: 3 × 10,000 = 30,000 cm².
  2. The trapezium formula is A = ½(a+b)h, not a×b×h: A = ½(6+10)(4) = 32 cm².
  3. The full circle area must be multiplied by the angle fraction (90/360 = ¼): A = ¼ × π × 36 = 28.3 cm², not the full circle area.

6. Mixed Practice — Bringing It All Together

  1. Convert 3.8 km to m.
  2. Find the area of a rectangle 11 cm by 4 cm.
  3. Find the area of a trapezium with parallel sides 5 cm and 9 cm, height 6 cm.
  4. Find the circumference and area of a circle with radius 6 cm.
  5. Find the arc length and sector area for radius 15 cm, angle 120°.
Show answers
  1. 3800 m
  2. 44 cm²
  3. 42 cm²
  4. C=37.7 cm, A=113.1 cm²
  5. Arc=(120/360)×2π(15)=31.4 cm. Sector area=(120/360)×π(15)²=235.6 cm²

Practice at Home

A large set to open the Mensuration unit — take it steadily.

Part A — Unit conversions
  1. Convert 7250 m to km.
  2. Convert 4.5 m² to cm².
  3. Convert 850,000 mm³ to cm³.
  4. Convert 1.8 litres to ml.
Part B — Area and perimeter
  1. Find the area and perimeter of a rectangle 12 cm by 7 cm.
  2. Find the area of a triangle with base 16 cm and height 5 cm.
  3. Find the area of a parallelogram with base 9 cm and height 6.5 cm.
  4. Find the area of a trapezium with parallel sides 7 cm and 13 cm, height 8 cm.
Part C — Circles
  1. Find the circumference of a circle with radius 9 cm.
  2. Find the area of a circle with diameter 20 cm.
  3. A circle has circumference 62.8 cm. Find its radius.
Part D — Arcs and sectors
  1. Find the arc length for radius 10 cm, angle 72°.
  2. Find the sector area for radius 14 cm, angle 45°.
  3. Find the arc length for radius 6 cm, angle 250°. Ext.
  4. A sector’s minor angle is 96°, radius 7 cm. Find the major sector’s arc length. Ext.
Part E — Applied / challenge
  1. A rectangular garden is 12 m by 8 m. A path 1.5 m wide runs around the inside edge of the garden. Find the area of the path.
  2. A pizza has radius 15 cm and is cut into 6 equal sectors. Find the area of one slice.
  3. A running track is made from a rectangle 80 m by 60 m with a semicircle (radius 30 m) attached to each of the two shorter ends. Find the total perimeter of the track.

Key Vocabulary Recap

Perimeter — the total distance around the outside of a shape
Trapezium — a quadrilateral with exactly one pair of parallel sides
Sector — a “slice” of a circle, bounded by two radii and an arc
Minor/major sector — the smaller/larger of the two sectors formed by a given angle Ext.

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C5.1, C5.2 and C5.3 (Core subject content) and sections E5.1, E5.2 and E5.3 (Extended subject content) — Units of measure, Area and perimeter, and Circles, arcs and sectors. Core restricts arc length/sector area problems to angles that are factors of 360°; Extended allows any angle and explicitly covers major sectors, marked above. Surface area and volume of solids, and compound shapes, are covered in a separate note.

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