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Mensuration

Mathematics · WAEC and JAMB · SS2 and SS3

Mensuration carries heavy Paper 2 marks and connects directly to everyday work: roofing, tanks, cement and land. The formulae are given to you in the mind, not on the paper, so they must be memorised exactly, including which ones use the slant height.

What you need to know

  • Perimeter and circumference are measured in single units such as cm, area in square units, and volume in cubic units. If your answer carries the wrong unit, you lose the accuracy mark.
  • The circumference of a circle is 2 pi r and the area is pi r^2. When the diameter is given, halve it first; mixing up r and d is the most expensive single error in this topic.
  • Use pi = 22/7 when the radius or diameter is a multiple of 7, because the sevens cancel and the arithmetic stays exact. Use 3.142 otherwise, or whichever value the question states.
  • For a sector of angle theta, the arc length is (theta/360) x 2 pi r and the area is (theta/360) x pi r^2. The fraction theta/360 is simply the share of the whole circle.
  • The area of a trapezium is half the sum of the parallel sides times the perpendicular height, written (1/2)(a + b)h. The height must be perpendicular, not a slanting side.
  • For any prism, including a cylinder, volume = cross-sectional area x length. Recognising a shape as a prism removes the need to memorise a separate formula.
  • The curved surface area of a cylinder is 2 pi r h, and the total surface area of a CLOSED cylinder adds the two circular ends, giving 2 pi r h + 2 pi r^2. An open tank has only one end.
  • For a cone, the curved surface area uses the SLANT height l, giving pi r l, while the volume uses the PERPENDICULAR height h, giving (1/3) pi r^2 h. The two heights are linked by l^2 = r^2 + h^2.
  • A sphere has surface area 4 pi r^2 and volume (4/3) pi r^3. A hemisphere has half that volume, and its total surface area is 2 pi r^2 plus the flat circle pi r^2, giving 3 pi r^2.
  • The volume of a pyramid is (1/3) x base area x perpendicular height, whatever the shape of the base. The one-third factor applies to every pointed solid.
  • One cubic metre holds 1,000 litres, and one cubic centimetre is one millilitre. Tank capacity questions nearly always need this conversion.
  • For composite solids, split the shape into standard parts, compute each separately, then add volumes or add only the EXPOSED surfaces. A cylinder topped by a hemisphere has no flat circle showing at the join.
  • The surface area of a solid cut open or hollowed includes the new faces created by the cut, which candidates regularly leave out.
  • When a solid is melted and recast, the VOLUME stays the same while the surface area changes. Set the two volume expressions equal and solve for the unknown dimension.

Key terms

Perimeter
The total distance round the outside boundary of a plane shape.
Sector
The region of a circle bounded by two radii and the arc between them.
Prism
A solid with a uniform cross-section along its whole length.
Slant height
The distance from the apex of a cone to a point on the edge of its base, measured along the sloping surface.
Total surface area
The combined area of every exposed face of a solid, including the curved surface and all flat ends.
Capacity
The volume a container can hold, usually expressed in litres rather than cubic units.

Formulae

  • circumference of circle = 2 * pi * r; area = pi * r^2
  • arc length = (theta/360) * 2 * pi * r; sector area = (theta/360) * pi * r^2
  • area of trapezium = (1/2)(a + b)h
  • volume of cylinder = pi * r^2 * h; curved surface area = 2 * pi * r * h; total surface area (closed) = 2 * pi * r(r + h)
  • volume of cone = (1/3) * pi * r^2 * h; curved surface area = pi * r * l; total surface area = pi * r(r + l)
  • cone relation: l^2 = r^2 + h^2
  • volume of sphere = (4/3) * pi * r^3; surface area = 4 * pi * r^2
  • volume of pyramid = (1/3) * base area * perpendicular height
  • 1 cubic metre = 1000 litres; 1 cubic centimetre = 1 millilitre

Worked examples

A cylindrical water tank has a radius of 1.5 m and a height of 2.8 m. Taking pi as 22/7, find its volume in cubic metres and its capacity in litres.

  1. Volume of a cylinder = pi r^2 h = (22/7) x (1.5)^2 x 2.8.
  2. First square the radius: 1.5^2 = 2.25.
  3. Simplify using the height: 2.8/7 = 0.4, so the expression becomes 22 x 0.4 x 2.25.
  4. 22 x 0.4 = 8.8, and 8.8 x 2.25 = 19.8.
  5. So the volume is 19.8 cubic metres.
  6. Convert to litres: 1 cubic metre = 1,000 litres, so 19.8 x 1,000 = 19,800 litres.

A solid cone has base radius 7 cm and slant height 25 cm. Taking pi as 22/7, find its total surface area and its volume.

  1. Total surface area = pi r(r + l) = (22/7) x 7 x (7 + 25).
  2. The 7 cancels with the denominator: 22 x 32 = 704, so the total surface area is 704 cm^2.
  3. For the volume, first find the perpendicular height using l^2 = r^2 + h^2: 25^2 = 7^2 + h^2.
  4. So h^2 = 625 - 49 = 576, giving h = 24 cm.
  5. Volume = (1/3) pi r^2 h = (1/3) x (22/7) x 49 x 24.
  6. (22/7) x 49 = 154, and 154 x 24 = 3,696.
  7. Divide by 3: 3,696/3 = 1,232 cm^3.

A sector of a circle of radius 10.5 cm subtends an angle of 120 degrees at the centre. Taking pi as 22/7, find the arc length and the area of the sector.

  1. The fraction of the circle is 120/360 = 1/3.
  2. Full circumference = 2 x (22/7) x 10.5. Since (22/7) x 10.5 = 33, the circumference is 66 cm.
  3. Arc length = (1/3) x 66 = 22 cm.
  4. Full area = pi r^2 = (22/7) x 10.5 x 10.5 = 33 x 10.5 = 346.5 cm^2.
  5. Sector area = (1/3) x 346.5 = 115.5 cm^2.

The mistake to avoid

Using the slant height in the cone volume formula, or the perpendicular height in the curved surface area, is the classic mensuration disaster. Equally common is forgetting that a tank, bucket or pipe is OPEN at the top, so its surface area must not include both circular ends.

In the exam

Write out the formula first, then substitute, because the formula line carries a mark on its own. Check whether the question gives you radius or diameter, and whether the solid is open or closed, before calculating anything. Use pi = 22/7 whenever the radius is a multiple of 7 or 3.5, and keep the unit, squared or cubed, on every line of working.