Surface area and volume of solids · 立体图形的表面积与体积
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| volume/ˈvɒljuːm/ | 体积 | tǐ jī |
| surface area/ˈsɜːfɪs ˈeərɪə/ | 表面积 | biǎo miàn jī |
| cuboid/ˈkjuːbɔɪd/ | 长方体 | cháng fāng tǐ |
| prism/ˈprɪzəm/ | 棱柱 | léng zhù |
| cylinder/ˈsɪlɪndə/ | 圆柱 | yuán zhù |
| cone/kəʊn/ | 圆锥 | yuán zhuī |
| sphere/sfɪə/ | 球 | qiú |
How much water fills a swimming pool?
- A pool 25 m long, 10 m wide, 2 m deep holds $25 \times 10 \times 2 = 500\text{ m}^3 = 500\,000$ litres.
- Volume 体积 tells you how much a solid can hold. Surface area 表面积 tells you how much material covers it.
Volume scaling lab · 体积缩放实验
surface area grows with scale^2 · 表面积随尺度^2增长
Change length scale and see volume grow faster than surface area. · 改变长度比例并观察体积比表面积增长更快。
Volume of common solids
| Solid | Volume |
|---|---|
| cuboid 长方体 | $l \times w \times h$ |
| prism 棱柱 | cross-section area $\times$ length |
| cylinder 圆柱 | $\pi r^2 h$ |
| pyramid | $\dfrac{1}{3} \times$ base area $\times h$ |
| cone 圆锥 | $\dfrac{1}{3}\pi r^2 h$ |
| sphere 球 | $\dfrac{4}{3}\pi r^3$ |
A prism has the same cross-section all along its length. A cylinder is a circular prism: cross-section $= \pi r^2$, so volume $= \pi r^2 \times h$.
The pyramids of Giza are giant square-based pyramids — real solids whose volume and surface area you can calculate
A cylinder has radius 5 cm and height 10 cm. Its volume is kπ cm³. What is k? · 一个圆柱体的半径为 5 cm，高为 10 cm。其体积为 kπ cm³。k 是多少？
Volume = πr²h = π(25)(10) = 250π, so k = 250. · 体积 = πr²h = π(25)(10) = 250π，所以 k = 250。
The volume of a sphere of radius r is: · 半径为 r 的球体体积为：
Sphere volume = (4/3)πr³; 4πr² is its surface area. · 球体体积 = (4/3)πr³；4πr² 是其表面积。
A cone has radius 3 cm and height 4 cm. Its volume is kπ cm³. What is k? · 一个圆锥的底面半径为 3 cm，高为 4 cm。其体积为 kπ cm³。k 是多少？
Volume = (1/3)πr²h = (1/3)π(9)(4) = 12π, so k = 12. · 体积 = (1/3)πr²h = (1/3)π(9)(4) = 12π，所以 k = 12。
A cuboid is 4 cm by 3 cm by 5 cm. Find its volume (cm³). · 一个长方体的尺寸为 4 cm × 3 cm × 5 cm。求其体积（cm³）。
Volume = l × w × h = 4 × 3 × 5 = 60 cm³. · 体积 = 长 × 宽 × 高 = 4 × 3 × 5 = 60 cm³。
Surface area
- Add the area of every outside face.
- Cylinder: curved surface $2\pi rh$ plus two circular ends $2\pi r^2$. Total $= 2\pi rh + 2\pi r^2$.
- Sphere: $4\pi r^2$.
Surface area ≠ volume. A cylinder's volume uses $\pi r^2 h$ (cubic units); its surface area uses $2\pi rh + 2\pi r^2$ (square units). Mixing them up is a common mistake.
The pyramids of Giza are square-based pyramids, a 3-D solid
The same cylinder (r = 5, h = 10) has total surface area kπ cm². What is k? · 同一个圆柱体（r = 5, h = 10）的总表面积为 kπ cm²。k 是多少？
2π(5)(10) + 2π(5²) = 100π + 50π = 150π, so k = 150. · 2π(5)(10) + 2π(5²) = 100π + 50π = 150π，所以 k = 150。
Surface area and volume of a cylinder use the same formula. · 圆柱体的表面积和体积使用相同的公式。
Volume = πr²h (cubic units); surface area = 2πrh + 2πr² (square units). Different formulas. · 体积 = πr²h（立方单位）；表面积 = 2πrh + 2πr²（平方单位）。公式不同。
Worked example
- Cylinder $r = 5$, $h = 10$:
- Volume $= \pi(5^2)(10) = 250\pi\text{ cm}^3$.
- Surface area $= 2\pi(5)(10) + 2\pi(5^2) = 100\pi + 50\pi = 150\pi\text{ cm}^2$.
Volume builds on area: a cylinder is a circle ($\pi r^2$) extended through a height ($h$).
Stacked containers are cuboids; volume is length times width times height
Cone area and pyramid volume
- A cone with radius 3 cm, perpendicular height 4 cm and slant height 5 cm has curved area $S_c=\pi rl=15\pi\text{ cm}^2$. Add its circular base for total $S=15\pi+9\pi=24\pi\text{ cm}^2$; use perpendicular height for volume $V=\pi r^2h/3=12\pi\text{ cm}^3$.
- A pyramid with base area $30\text{ cm}^2$ and perpendicular height 8 cm has $V=Ah/3=80\text{ cm}^3$. For its surface area, add the base and the triangular faces, using each face's perpendicular height.
A cone has r = 3 cm and slant height 5 cm. Its total surface area is kπ cm². Find k. · 一个圆锥的底面半径 r = 3 cm，母线长 ⟨slant height⟩ 为5 cm。其总表面积为 kπ cm²。求 k 的值。
Curved area is 15π and base area is 9π; k = 24. · 侧面积为15π，底面积为9π；因此 k = 24。
A pyramid has base area 30 cm² and perpendicular height 8 cm. Find its volume in cm³. · 一个棱锥的底面积为30 cm²，高为8 cm。求其体积，单位为cm³。
V = base area × height / 3 = 30 × 8 / 3 = 80. · V = 底面积 × 高 / 3 = 30 × 8 / 3 = 80。
You've got it
- cylinder volume $= \pi r^2 h$; sphere $= \dfrac{4}{3}\pi r^3$; cone $= \dfrac{1}{3}\pi r^2 h$
- cylinder surface $= 2\pi rh + 2\pi r^2$; sphere surface $= 4\pi r^2$
- $r=5$, $h=10$ cylinder: volume $250\pi$, surface $150\pi$