Shapes, solids and units · 平面、立体与单位
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| area/ˈeərɪə/ | 面积 | miàn jī |
| volume/ˈvɒljuːm/ | 体积 | tǐ jī |
| units/ˈjuːnɪts/ | 单位 | dān wèi |
| polygon/ˈpɒlɪɡən/ | 多边形 | duō biān xíng |
| congruent/ˈkɒŋɡruːənt/ | 全等 | quán děng |
| similar/ˈsɪmɪlə/ | 相似 | xiāng sì |
| Pythagoras' theorem/paɪˈθæɡərəs ˈθɪərəm/ | 勾股定理 | gōu gǔ dìng lǐ |
| hypotenuse/haɪˈpɒtənjuːs/ | 斜边 | xié biān |
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? · 六边形的内角和是多少度?
(n − 2) × 180° with n = 6 gives 4 × 180° = 720°. · 用 (n − 2) × 180°,n = 6,得 4 × 180° = 720°。
Two congruent shapes are always similar. · 两个全等的图形一定相似。
Congruent means identical, which is similarity with a ratio of 1. The reverse is not true. · 全等意味着完全相同,相当于比例为 1 的相似。反过来则不成立。
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? · 直角三角形的两条直角边为 9 cm 与 12 cm,斜边有多长(cm)?
81 + 144 = 225, and √225 = 15. It is the 3-4-5 triangle scaled by 3. · 81 + 144 = 225,√225 = 15。这是 3-4-5 三角形放大三倍。
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. · 一个圆柱半径 1.2 米、高 3.0 米,求其体积(立方米,保留两位小数)。
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,约合 13 570 升。
How many litres are there in one cubic metre? · 一立方米等于多少升?
One cubic metre is 1000 litres. This single conversion accounts for most unit errors in the module. · 一立方米等于 1000 升。本模块中大多数单位错误都出在这一个换算上。
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? · 把一个立体的所有长度加倍,它的体积会怎样变化?
Volume scales with the cube of the length ratio: 2³ = 8. Area scales with the square, 4. · 体积按长度比的三次方缩放:2³ = 8。面积按平方缩放,是 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.