Shapes, solids and units · Figuras, sólidos y unidades
| English | Español |
|---|---|
| area/ˈeərɪə/ | área |
| volume/ˈvɒljuːm/ | volumen |
| units/ˈjuːnɪts/ | unidades |
| polygon/ˈpɒlɪɡən/ | polígono |
| congruent/ˈkɒŋɡruːənt/ | congruentes |
| similar/ˈsɪmɪlə/ | similar |
| Pythagoras' theorem/paɪˈθæɡərəs ˈθɪərəm/ | Teorema de Pitágoras |
| hypotenuse/haɪˈpɒtənjuːs/ | hipotenusa |
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? · ¿Cuánto suman los ángulos interiores de un hexágono, en grados?
(n − 2) × 180° with n = 6 gives 4 × 180° = 720°. · (n − 2) × 180° con n = 6 da 4 × 180° = 720°.
Two congruent shapes are always similar. · Dos figuras congruentes siempre son similares.
Congruent means identical, which is similarity with a ratio of 1. The reverse is not true. · Congruente significa idéntico, lo cual es similitud con una razón de 1. Lo contrario no es cierto.
Choose the perpendicular height
- Pythagoras' theorem · Teorema de Pitágoras 勾股定理 uses · usa $c^2=a^2+b^2$ 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 triángulo rectángulo tiene catetos de 9 cm y 12 cm. ¿Cuál es la longitud de la hipotenusa, en cm?
81 + 144 = 225, and √225 = 15. It is the 3-4-5 triangle scaled by 3. · 81 + 144 = 225, y √225 = 15. Es el triángulo 3-4-5 escalado por 3.
Scaling changes different dimensions differently
- A length factor k gives area · área 面积 factor $k^2$ and · y volume · volumen 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 cilindro tiene radio 1.2 m y altura 3.0 m. Indique su volumen en metros cúbicos, con 2 decimales.
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, que equivale a aproximadamente 13 570 litros.
How many litres are there in one cubic metre? · ¿Cuántos litros hay en un metro cúbico?
One cubic metre is 1000 litres. This single conversion accounts for most unit errors in the module. · Un metro cúbico son 1000 litros. Esta única conversión explica la mayoría de los errores de unidades en el módulo.
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? · Se duplica cada longitud de un sólido. ¿Qué sucede con su volumen?
Volume scales with the cube of the length ratio: 2³ = 8. Area scales with the square, 4. · El volumen escala con el cubo de la razón de longitudes: 2³ = 8. El área escala con el cuadrado, 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.