Shapes, solids and units
| English | Chinese | Pinyin |
|---|---|---|
| volume | 体积 | tǐ jī |
| units | 单位 | dān wèi |
| polygon | 多边形 | duō biān xíng |
| congruent | 全等 | quán děng |
| similar | 相似 | xiāng sì |
| Pythagoras' theorem | 勾股定理 | gōu gǔ dìng lǐ |
| hypotenuse | 斜边 | xié biān |
| area | 面积 | miàn jī |
The shape is not the question
- A cylinder question rarely asks "what is a cylinder". It asks for a volume in litres, from a radius in metres.
- The marks are in the units 单位, the formula choice, and the reasoning — not in the shape.
- Unit 4 is where careless unit conversion costs most marks in the whole module.
Angles you can rely on
- Angles on a straight line add to $180°$; around a point, $360°$; in a triangle, $180°$.
- In an $n$-sided polygon 多边形 the interior angles add to $(n-2) \times 180°$.
- Congruent 全等 shapes are identical; similar 相似 shapes have the same shape with lengths in one ratio.
What do the interior angles of a hexagon add up to, in degrees?
(n − 2) × 180° with n = 6 gives 4 × 180° = 720°.
Two congruent shapes are always similar.
Congruent means identical, which is similarity with a ratio of 1. The reverse is not true.
Pythagoras and the right angle
- Pythagoras' theorem 勾股定理: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse 斜边.
- It works only in a right-angled triangle — check for the right angle before you use it.
- It is also the distance formula in disguise, which is why unit 3 and unit 4 keep meeting.
A right-angled triangle has legs 9 cm and 12 cm. How long is the hypotenuse, in cm?
81 + 144 = 225, and √225 = 15. It is the 3-4-5 triangle scaled by 3.
A cylindrical tank has radius 1.2 m and height 3.0 m. How many litres does it hold?
$V = \pi r^2 h = \pi \times 1.2^2 \times 3.0 = 13.57\ \text{m}^3$
$1\ \text{m}^3 = 1000$ litres, so the tank holds about $13\,570$ litres.
Carrying the unit through every line is what makes the final conversion obvious — and what earns the method mark even if the arithmetic slips.
A cylinder has radius 1.2 m and height 3.0 m. Give its volume in cubic metres, to 2 decimal places.
V = πr²h = π × 1.44 × 3.0 = 13.57 m³, which is about 13 570 litres.
How many litres are there in one cubic metre?
One cubic metre is 1000 litres. This single conversion accounts for most unit errors in the module.
Scaling is not linear
- Double every length of a solid and its area 面积 multiplies by $2^2 = 4$.
- Its volume 体积 multiplies by $2^3 = 8$.
- This is why a scale model that looks twice as big weighs eight times as much.
Every length of a solid is doubled. What happens to its volume?
Volume scales with the cube of the length ratio: 2³ = 8. Area scales with the square, 4.
Write the unit on every line, not only on the answer. A calculation in metres that ends in litres has a conversion in it somewhere, and the line where the unit changes is the line to check.
$\pi r^2$ is an area and $2\pi r$ is a circumference. Students who memorise the symbols rather than the meaning use the wrong one under time pressure. A quick test: an area formula must have a squared length in it.