5 Geometry and trigonometry

Learning objectives

55C Areas, surface areas and volumesCalculate areas of triangles and rectangles, and surface areas and volumes of cubes.

Length is one-dimensional, area measures a two-dimensional surface, and volume measures three-dimensional space. Convert all lengths to the same unit before applying a formula, then square or cube that unit in the result.

Shape or quantity	Formula	Example
rectangle area	$A=lw$	$8\text{ cm}\times3\text{ cm}=24\text{ cm}^2$
triangle area	$A=\tfrac12 bh$ using perpendicular height	$\tfrac12(6\text{ cm})(4\text{ cm})=12\text{ cm}^2$
cube surface area	$S=6s^2$ because a cube has six square faces	for $s=4\,\mu\mathrm{m}$ , $S=6(4^2)=96\,\mu\mathrm{m}^2$
cube volume	$V=s^3$	for $s=4\,\mu\mathrm{m}$ , $V=4^3=64\,\mu\mathrm{m}^3$

Label the required dimension, select the matching formula, substitute one consistent length unit, calculate, and check that the final unit is squared for area or cubed for volume. Surface-area-to-volume comparisons must use compatible units before division.

The sloping side of a triangle is not its height unless it is perpendicular to the chosen base. For a cube, $s^2$ is the area of one face; total surface area is $6s^2$ , while volume is $s^3$ .

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