Measurement — CSEC Maths notes

Units and conversion

$1\,\text{m} = 100\,\text{cm}$ and $1\,\text{cm} = 10\,\text{mm}$ .

Area and volume conversions are not the same factor:

$1\,\text{m}^2 = 100 \times 100 = 10\,000\,\text{cm}^2 \qquad 1\,\text{m}^3 = 100^3 = 1\,000\,000\,\text{cm}^3$

Also worth knowing: $1$ litre $= 1000\,\text{cm}^3$ , and $1$ tonne $= 1000\,\text{kg}$ .

Perimeter and area

Shape	Area
rectangle	$l \times w$
triangle	$\tfrac12 \times \text{base} \times \text{height}$
parallelogram	$\text{base} \times \text{height}$
trapezium	$\tfrac12(a + b)h$ , where $a$ and $b$ are the parallel sides

Perimeter is the distance round the outside; area is the space inside. Perimeter is measured in cm, area in cm².

For an awkward shape, cut it into rectangles and triangles, work out each piece, and add.

Circles

For a circle of radius 7 cm with $\pi = \tfrac{22}{7}$ : the circumference is $2 \times \tfrac{22}{7} \times 7 = 44$ cm, and the area is $\tfrac{22}{7} \times 49 = 154$ cm².

Arcs and sectors

A sector is a "slice" of a circle. Everything about it is the same fraction of the whole circle as its angle is of $360°$ .

There is really only one idea here: find the fraction $\dfrac{\theta}{360}$ , then apply it to the whole circumference or the whole area.

Volume and surface area

Solid	Volume	Surface area
cuboid	$l \times w \times h$	$2(lw + lh + wh)$
prism	cross-section area $\times$ length
cylinder	$\pi r^2 h$	curved $2\pi r h$ ; total $2\pi r h + 2\pi r^2$
cone	$\tfrac13 \pi r^2 h$	curved $\pi r l$
sphere	$\tfrac43 \pi r^3$	$4\pi r^2$

A cylinder is just a prism whose cross-section is a circle, which is why its volume is $\pi r^2 \times h$ — the area of the circular face times the length.

Working backwards is common too. If a cube has volume $64$ cm³, its edge is $\sqrt[3]{64} = 4$ cm — a cube root, because the volume was the edge cubed.