Shapes, solids and units · Formes, solides et unités
| English | Français |
|---|---|
| area/ˈeərɪə/ | aire |
| volume/ˈvɒljuːm/ | volume |
| units/ˈjuːnɪts/ | unités |
| polygon/ˈpɒlɪɡən/ | polygone |
| congruent/ˈkɒŋɡruːənt/ | congruentes |
| similar/ˈsɪmɪlə/ | semblables |
| Pythagoras' theorem/paɪˈθæɡərəs ˈθɪərəm/ | Théorème de Pythagore |
| hypotenuse/haɪˈpɒtənjuːs/ | hypoténuse |
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? · Quelle est la somme des angles intérieurs d'un hexagone, en degrés ?
(n − 2) × 180° with n = 6 gives 4 × 180° = 720°. · (n − 2) × 180° avec n = 6 donne 4 × 180° = 720°.
Two congruent shapes are always similar. · Deux formes congruentes sont toujours semblables.
Congruent means identical, which is similarity with a ratio of 1. The reverse is not true. · Congruent signifie identique, ce qui est une similarité avec un rapport de 1. L'inverse n'est pas vrai.
Choose the perpendicular height
- Pythagoras' theorem · Théorème de Pythagore 勾股定理 uses · utilise $c^2=a^2+b^2$ 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? · Un triangle rectangle a des côtés de l'angle droit de 9 cm et 12 cm. Quelle est la longueur de l'hypoténuse, en cm ?
81 + 144 = 225, and √225 = 15. It is the 3-4-5 triangle scaled by 3. · 81 + 144 = 225, et √225 = 15. C'est le triangle 3-4-5 mis à l'échelle par 3.
Scaling changes different dimensions differently
- A length factor k gives area · aire 面积 factor $k^2$ and · et 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. · Un cylindre a un rayon de 1.2 m et une hauteur de 3.0 m. Donner son volume en mètres cubes, arrondi à 2 décimales.
V = πr²h = π × 1.44 × 3.0 = 13.57 m³, which is about 13 570 litres. · V = πr²h = π × 1.44 × 3.0 = 13.57 m³, ce qui équivaut à environ 13 570 litres.
How many litres are there in one cubic metre? · Combien de litres y a-t-il dans un mètre cube ?
One cubic metre is 1000 litres. This single conversion accounts for most unit errors in the module. · Un mètre cube équivaut à 1000 litres. Cette seule conversion explique la plupart des erreurs d'unités dans ce 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? · Toute longueur d'un solide est doublée. Que devient son volume ?
Volume scales with the cube of the length ratio: 2³ = 8. Area scales with the square, 4. · Le volume évolue selon le cube du rapport de longueur : 2³ = 8. L'aire évolue selon le carré, 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.