Mensuration is the section of the Edexcel IGCSE Mathematics Specification B exam where you calculate lengths, areas, and volumes. The questions are direct, the formulae are given on the formula sheet, and the marks go to students who set up the calculation correctly and execute it without error.
The edexcel igcse mathematics specification b mensuration content sits under a single heading, "Length, area and volume," but the range is wide. You need to handle 2D shapes (rectangles, triangles, parallelograms, trapeziums, circles, composite figures), 3D shapes (cuboids, prisms, cylinders, cones, spheres, pyramids, composite solids), and circular measures (arc length, sector area). The formula sheet provided in the exam covers the less common formulae, but you should know the standard ones from memory so you do not waste time looking them up.
Areas of 2D shapes
Standard formulae
| Shape | Area formula |
|---|---|
| Rectangle | length x width |
| Triangle | (1/2) x base x height |
| Parallelogram | base x perpendicular height |
| Trapezium | (1/2)(a + b) x h, where a and b are the parallel sides |
| Circle | pi x r2 |
Worked example: A trapezium has parallel sides of 8 cm and 14 cm and a perpendicular height of 5 cm. Find its area.
Area = (1/2)(8 + 14) x 5 = (1/2)(22)(5) = 55 cm2.
Composite shapes
Composite shapes are formed by combining or removing standard shapes. The mensuration edexcel igcse exam frequently presents L-shapes, shapes with semicircular ends, or regions between two circles (annuli).
Worked example: A running track consists of a rectangle 100 m by 60 m with a semicircle at each short end. Find the total area.
Rectangle area = 100 x 60 = 6000 m2.
Each semicircle has diameter 60 m, so radius = 30 m.
Two semicircles = one full circle: area = pi x 302 = 900pi m2.
Total area = 6000 + 900pi = 6000 + 2827.4... = 8827 m2 (to 4 s.f.).
Volumes of 3D shapes
Standard formulae
| Shape | Volume formula |
|---|---|
| Cuboid | length x width x height |
| Prism | cross-sectional area x length |
| Cylinder | pi x r2 x h |
| Cone | (1/3) x pi x r2 x h |
| Sphere | (4/3) x pi x r3 |
| Pyramid | (1/3) x base area x height |
The cone, sphere, and pyramid formulae are on the formula sheet. The cuboid, prism, and cylinder formulae are not, and you must know them.
Worked example: A cone has radius 6 cm and height 10 cm. Find its volume.
V = (1/3) x pi x 62 x 10 = (1/3) x pi x 360 = 120pi = 376.99... = 377.0 cm3 (to 4 s.f.).
Worked example: A sphere has volume 500 cm3. Find its radius.
500 = (4/3) x pi x r3.
r3 = 500 x 3 / (4pi) = 1500 / (4pi) = 375/pi.
r3 = 119.366...
r = the cube root of 119.366... = 4.924... = 4.92 cm (to 3 s.f.).
Composite solids
The igcse 4mb1 mensuration questions often involve solids made from two or more basic shapes. A hemisphere on top of a cylinder, a cone removed from a cylinder, or a pyramid on top of a cuboid.
Worked example: A solid consists of a cylinder of radius 5 cm and height 8 cm with a hemisphere of radius 5 cm on top. Find the total volume.
Cylinder volume = pi x 25 x 8 = 200pi cm3.
Hemisphere volume = (1/2) x (4/3) x pi x 125 = (2/3) x 125pi = 250pi/3 cm3.
Total = 200pi + 250pi/3 = 600pi/3 + 250pi/3 = 850pi/3 = 890.1 cm3 (to 4 s.f.).
Arc length and sector area
A sector is a "slice" of a circle, defined by two radii and the arc between them. The angle at the centre determines what fraction of the full circle the sector represents.
Arc length = (theta/360) x 2pi r.
Sector area = (theta/360) x pi r2.
where theta is the angle at the centre in degrees.
Worked example: A sector has radius 12 cm and angle 75 degrees. Find the arc length and the area.
Arc length = (75/360) x 2pi x 12 = (75/360) x 24pi = (5/24) x 24pi = 5pi = 15.71 cm (to 4 s.f.).
Sector area = (75/360) x pi x 144 = (75/360) x 144pi = 30pi = 94.25 cm2 (to 4 s.f.).
Worked example: The area of a sector is 50 cm2 and the radius is 10 cm. Find the angle of the sector.
50 = (theta/360) x pi x 100.
theta/360 = 50/(100pi) = 1/(2pi).
theta = 360/(2pi) = 180/pi = 57.3 degrees (to 3 s.f.).
Surface area
Surface area questions require you to find the total area of all external faces. The edexcel igcse mathematics specification b notes on surface area cover:
- Cylinder: 2pi r2 + 2pi rh (two circular ends plus the curved surface).
- Cone: pi r2 + pi rl (circular base plus curved surface, where l is the slant height).
- Sphere: 4pi r2.
Worked example: Find the total surface area of a cone with radius 5 cm and slant height 13 cm.
Base area = pi x 25 = 25pi.
Curved surface area = pi x 5 x 13 = 65pi.
Total = 25pi + 65pi = 90pi = 282.7 cm2 (to 4 s.f.).
Converting between units
Unit conversions are a frequent source of lost marks. The key relationships are:
- 1 m = 100 cm, so 1 m2 = 10,000 cm2 and 1 m3 = 1,000,000 cm3.
- 1 km = 1000 m, so 1 km2 = 1,000,000 m2.
- 1 litre = 1000 cm3 = 0.001 m3.
When a question gives dimensions in different units, convert everything to the same unit before substituting into a formula. A cylinder with radius 50 cm and height 2 m must use either r = 50 cm, h = 200 cm, or r = 0.5 m, h = 2 m. Mixing units produces an answer that is off by a factor of 100 or more.
Common mistakes in Mensuration
- Confusing radius and diameter. If the question gives a diameter of 10 cm, the radius is 5 cm. Substituting 10 into a formula that requires the radius quadruples the area and octuples the volume. This is the single most common error in this section.
- Forgetting units. Area is in square units (cm2), volume is in cubic units (cm3). Omitting or using the wrong unit loses the accuracy mark.
- Using the wrong height for cones and pyramids. The volume formula uses the perpendicular height, not the slant height. If only the slant height is given, use Pythagoras to find the perpendicular height first.
- Rounding too early. Keep full precision (or use pi) until the final answer. Rounding intermediate results in multi-step problems can shift the final answer outside the acceptable range.
- Missing faces in surface area. For a hemisphere, the surface area is 2pi r2 (curved) plus pi r2 (flat circular face) = 3pi r2. Omitting the flat face is a common oversight.
Self-check questions
Work through these edexcel igcse mathematics specification b practice questions fully before looking at the solutions.
- A prism has a cross-section that is a right-angled triangle with legs 6 cm and 8 cm. The prism is 15 cm long. Find the volume.
- A sector of a circle has radius 9 cm and arc length 12 cm. Find the angle of the sector and the area of the sector.
- A solid hemisphere has radius 7 cm. Find the total surface area (curved surface plus flat face).
- A cylindrical tank has radius 3 m and height 5 m. Water fills the tank to a depth of 4 m. Find the volume of water in the tank.
Solutions
1. Cross-sectional area = (1/2)(6)(8) = 24 cm2. Volume = 24 x 15 = 360 cm3.
2. Arc length = (theta/360) x 2pi r. 12 = (theta/360) x 18pi. theta = 12 x 360/(18pi) = 4320/(18pi) = 240/pi = 76.4 degrees (to 3 s.f.). Area = (76.39.../360) x pi x 81 = 54.0 cm2 (to 3 s.f.). Alternatively, area = (1/2) x r x arc length = (1/2)(9)(12) = 54 cm2 exactly.
3. Curved surface = 2pi x 49 = 98pi. Flat face = pi x 49 = 49pi. Total = 147pi = 461.8 cm2 (to 4 s.f.).
4. Volume = pi x 32 x 4 = 36pi = 113.1 m3 (to 4 s.f.).
Mensuration rewards the student who reads the question carefully, selects the right formula, identifies the correct dimensions, and executes the arithmetic without shortcuts. The edexcel igcse mathematics specification b revision notes for this section should be a compact reference of formulae, paired with enough worked examples that the method becomes automatic. Keep your edexcel igcse mathematics specification b explained working clear and labelled, and the marks will follow.
Edexcel IGCSE Mathematics Specification B Mensuration revision notes: area, volume, arcs, sectors, cones, spheres, and worked examples.
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