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7 Mensuration

Syllabus
2016
Topic
7
Level
—

Learning objectives

7A Length, area and volumeLength, area, and volume7B 2D mensurationMensuration of two-dimensional shapes, rectangle, parallelogram, trapezium, triangle, circle Straightforward calculations, where appropriate, of areas of the shapes mentioned and also of two-dimensional shapes that can be divided into a collection of such shapes (e.g. trapezia, polygons)7C 3D mensurationMensuration of three-dimensional shapes, right circular cylinder, right circular cone and sphere, cuboid, pyramid, prism Straightforward calculations, where appropriate, of volumes of the shapes mentioned and also of three-dimensional shapes which can be divided into a collection of such shapes (e.g. cone, hemisphere)7D Arcs and sectorsLength of an arc, area of a sector of a circle Radian measure is excluded

Track dimensions and units in mensuration

Length measures one dimension, area measures a surface in square units, and volume measures space in cubic units. Decide which quantity the question asks for before choosing a formula.

Quantity Example unit Conversion pattern
length cm\mathrm{cm} multiply the linear conversion factor
area cm2\mathrm{cm^2} square the linear conversion factor
volume cm3\mathrm{cm^3} cube the linear conversion factor

1\text{ m}=100\text{ cm}\Rightarrow 1\text{ m}^2=10,000\text{ cm}^2,\quad 1\text{ m}^3=1,000,000\text{ cm}^3

Convert all lengths to one unit before substitution and attach the correct power to the final unit. Perimeter is a length, not an area; surface area is not volume.

Find areas by selecting and combining shapes

Shape Area
rectangle lwlw
parallelogram bhbh
triangle frac12bhfrac12bh
trapezium frac12(a+b)hfrac12(a+b)h
circle πr2\pi r^2

The height is the perpendicular distance to the chosen base. A sloping side is not the height unless it is perpendicular to that base.

For a composite region, draw or imagine cuts into known shapes, calculate each part, then add included pieces or subtract holes. Check that the pieces cover the target once with no gaps or overlap.

Use radius, not diameter, in πr2\pi r^2. Keep π\pi exact until the final rounding step, and do not add areas expressed in different square units.

Calculate volumes of solids and composite solids

Solid Volume
prism cross-sectional area imesimes length
cuboid lwhlwh
cylinder πr2h\pi r^2h
pyramid frac13imesfrac13 imes base area imeshimes h
cone frac13πr2hfrac13\pi r^2h
sphere frac43πr3frac43\pi r^3

For a cone or pyramid, hh is the perpendicular height from the base to the apex, not a sloping edge. For a prism, its length runs perpendicular to the constant cross-section.

Split a compound solid into standard solids or subtract a drilled-out part. A hemisphere has half the volume of a sphere; match shared radii and heights before combining expressions.

Do not confuse curved surface features with volume. Convert every length first, cube the resulting unit, and delay decimal rounding until the complete volume has been evaluated.

Use degree fractions for arcs and sectors

\text{arc length}=\frac{\theta}{360^\circ}\times2\pi r

\text{sector area}=\frac{\theta}{360^\circ}\times\pi r^2

Treat heta/360∘heta/360^\circ as the fraction of a full circle. For a perimeter, include any straight radii as well as the arc; for a shaded region, subtract inner sectors when necessary.

The syllabus uses degrees and excludes radian measure. Arc length has linear units, sector area has square units, and the major sector uses 360∘−heta360^\circ- heta when hetaheta labels the minor angle.