IGCSE Mathematics: Mensuration — Revision Notes

Condensed for the final weeks. Pair these notes with the Mensuration practice questions for worked exam-style application.

Units of measure

Linear conversions apply directly, but area conversions square the factor and volume conversions cube it – this is the most-tested single idea in the whole topic, and the source of more lost marks than any formula itself:

1 m = 100 cm            -> linear factor 100
1 m^2 = 100^2 = 10 000 cm^2   -> area factor 100^2
1 m^3 = 100^3 = 1 000 000 cm^3 -> volume factor 100^3

Forgetting to square or cube the linear factor is the single most common unit-conversion error in this topic, and it is worth double-checking on every question that involves converting an area or a volume, not just the ones that explicitly say “convert”.

Area and perimeter

Shape	Area	Perimeter
Rectangle	length x width	2(length + width)
Triangle	(1/2) x base x height	sum of the three sides
Parallelogram	base x height	sum of the four sides
Trapezium	(1/2)(a + b) x height, a and b the parallel sides	sum of the four sides

Triangle: base 9 cm, height 6 cm
Area = (1/2) x 9 x 6 = 27 cm^2

Trapezium: parallel sides 5 cm and 9 cm, height 4 cm
Area = (1/2)(5 + 9) x 4 = (1/2)(14)(4) = 28 cm^2

Circles, arcs and sectors

circumference = 2 x pi x r          area = pi x r^2

arc length  = (angle / 360) x 2 x pi x r
sector area = (angle / 360) x pi x r^2

Both the arc length and sector-area formulas scale the full circumference or area by the fraction of a full turn the angle represents. For a major sector or arc, first subtract the given (minor) angle from 360 degrees to get the correct larger angle before applying the fraction – applying the fraction directly to the given angle answers the wrong region.

Surface area and volume

Cylinder:  volume = pi x r^2 x h
           total surface area = 2 x pi x r x h + 2 x pi x r^2

Cone:      volume = (1/3) x pi x r^2 x h
           (h is the PERPENDICULAR height, not the slant height)

Sphere:    volume = (4/3) x pi x r^3
           surface area = 4 x pi x r^2

A cone’s slant height and perpendicular height are different lengths, related by Pythagoras’ theorem (slant height is the hypotenuse of a right-angled triangle formed with the radius and the perpendicular height) – always check which one a question has given before substituting into the volume formula. Total surface area of a cylinder or cone must include the circular end(s); curved surface area alone does not.

Cone: radius 3 cm, slant height 5 cm
perpendicular height: h^2 = 5^2 - 3^2 = 25 - 9 = 16 -> h = 4 cm
volume = (1/3) x pi x 3^2 x 4 = 37.7 cm^3 (3 s.f.)

Compound shapes and parts of shapes

A compound shape’s area or volume is found by adding or subtracting the areas or volumes of the simpler shapes it is built from – decide which operation applies by checking whether a piece has been joined on or removed.

10 cm x 6 cm rectangle, semicircle of diameter 6 cm removed:
rectangle area  = 10 x 6 = 60 cm^2
semicircle area = (1/2) x pi x 3^2 = 14.14 cm^2
remaining area  = 60 - 14.14 = 45.9 cm^2 (3 s.f.)

The same add-or-subtract principle applies to compound solids as to compound areas – a solid made of a cylinder with a cone removed from one end uses the same subtraction logic as the 2D rectangle-minus-semicircle example above, just with volume formulas in place of area formulas.

Exam traps

• Converting an area or volume using only the linear conversion factor, without squaring or cubing it.
• Forgetting the angle-over-360 fraction when finding an arc length or sector area.
• Using the slant height instead of the perpendicular height in a cone’s volume formula.
• Applying the minor-sector angle directly to a major-sector question, instead of subtracting it from 360 degrees first.
• Adding instead of subtracting (or the reverse) when a compound shape has a piece removed.
• Rounding at an intermediate step in a multi-step calculation rather than only at the final answer.

Self-test

1. State the area-conversion factor between m^2 and cm^2.
2. Write down the formula for the area of a sector, given the sector angle and radius.
3. Which length is used in a cone’s volume formula: the slant height or the perpendicular height?
4. A major sector has a minor angle of 80 degrees marked on the diagram. What angle should be used to find the major sector’s area?
5. State whether a compound-shape calculation with a piece removed uses addition or subtraction.
6. Write down the area formula for a trapezium with parallel sides a and b and height h.

Answers: 1. 10 000 (100^2), since 1 m = 100 cm. 2. Sector area = (angle / 360) x pi x r^2. 3. The perpendicular height. 4. 360 - 80 = 280 degrees. 5. Subtraction – subtract the removed piece’s area from the whole shape’s area. 6. Area = (1/2)(a + b) x h.