Shapes, solids and units
Introduced| English | Meaning |
|---|---|
| area/ˈeərɪə/ | The measure of a two-dimensional region, expressed in square units. |
| volume/ˈvɒljuːm/ | The amount of three-dimensional space occupied. |
| units/ˈjuːnɪts/ | Agreed reference quantities used to express measurements or counts. |
| polygon/ˈpɒlɪɡən/ | A closed plane shape whose boundary consists of straight line segments. |
| congruent/ˈkɒŋɡruːənt/ | Having exactly the same size and shape. |
| similar/ˈsɪmɪlə/ | Describing shapes with equal corresponding angles and proportional corresponding lengths. |
| Pythagoras' theorem/paɪˈθæɡərəs ˈθɪərəm/ | In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. |
| hypotenuse/haɪˈpɒtənjuːs/ | The side opposite the right angle in a right-angled triangle. |
Distinguish area from volume
- A container's capacity depends on its internal dimensions. A calculation using metres needs an explicit conversion before an answer in litres.
- Units 单位 make the distinction visible: a length uses cm, an area uses square centimetres, and a volume uses cubic centimetres.
State the geometry assumptions
- In plane Euclidean geometry, straight-line angles sum to 180 degrees and a triangle's angles to 180 degrees. A simple n-sided polygon 多边形 has interior-angle sum $(n-2)180^\circ$.
- Congruent 全等 shapes have the same shape and size; similar 相似 shapes have equal corresponding angles and proportional corresponding lengths.
What do the interior angles of a hexagon add up to, in degrees?
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(n − 2) × 180° with n = 6 gives 4 × 180° = 720°.
Two congruent shapes are always similar.
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Congruent means identical, which is similarity with a ratio of 1. The reverse is not true.
Choose the perpendicular height
- Pythagoras' theorem 勾股定理 uses $c^2=a^2+b^2$ only for a right triangle. Its hypotenuse 斜边 is opposite the right angle.
- Triangle area is $A=bh/2$, with h perpendicular to the base. Prism volume is cross-sectional area times length; a sloping edge is not automatically the height.
A right-angled triangle has legs 9 cm and 12 cm. How long is the hypotenuse, in cm?
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81 + 144 = 225, and √225 = 15. It is the 3-4-5 triangle scaled by 3.
Scaling changes different dimensions differently
- A length factor k gives area 面积 factor $k^2$ and volume 体积 factor $k^3$ for geometrically similar objects.
- Doubling all three dimensions multiplies volume by 8. Mass grows by that factor only if density and the compared occupied-volume assumptions also match.
A cylinder has radius 1.2 m and height 3.0 m. Give its volume in cubic metres, to 2 decimal places.
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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?
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One cubic metre is 1000 litres. This single conversion accounts for most unit errors in the module.
A triangular prism. A right-triangle cross-section has perpendicular sides 3 and 4 cm, so $A=bh/2=6$ square centimetres. With prism length 10 cm, $V=AL=60$ cubic centimetres. Each formula measures a different part of the solid.
Every length of a solid is doubled. What happens to its volume?
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Volume scales with the cube of the length ratio: 2³ = 8. Area scales with the square, 4.
Surface area is a sum of faces. For a closed rectangular block, $S=2(lw+lh+wh)$, while $V=lwh$. Sheet 1.4 includes capacity and displacement, with complete submersion and no spilling stated explicitly.
A submerged object displaces its submerged volume, not its mass. Divide displaced volume by the tank's horizontal cross-sectional area to obtain the rise. Check whether water would overflow before using a no-spill answer.