Surface area and volume of solids
Introduced| English | Meaning |
|---|---|
| volume/ˈvɒljuːm/ | The amount of three-dimensional space occupied. |
| surface area/ˈsɜːfɪs ˈeərɪə/ | The total area of the outer surfaces of a three-dimensional object. |
| cuboid/ˈkjuːbɔɪd/ | A solid with six rectangular faces meeting at right angles. |
| prism/ˈprɪzəm/ | A solid with two congruent, parallel polygonal bases joined by parallelogram faces. |
| cylinder/ˈsɪlɪndə/ | Here, a right circular solid with two equal parallel circular ends, an axis perpendicular to them and a curved side. |
| cone/kəʊn/ | A solid with a circular base and a curved surface that narrows to a vertex. |
| sphere/sfɪə/ | A three-dimensional shape whose surface points are all the same distance from its centre. |
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
surface area grows with scale^2
Change length scale and see volume grow faster than surface area.
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?
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Volume = πr²h = π(25)(10) = 250π, so k = 250.
The volume of a sphere of radius r is:
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Sphere volume = (4/3)πr³; 4πr² is its surface area.
A cone has radius 3 cm and height 4 cm. Its volume is kπ cm³. What is k?
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Volume = (1/3)πr²h = (1/3)π(9)(4) = 12π, so k = 12.
A cuboid is 4 cm by 3 cm by 5 cm. Find its volume (cm³).
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Volume = l × w × h = 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?
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2π(5)(10) + 2π(5²) = 100π + 50π = 150π, so k = 150.
Surface area and volume of a cylinder use the same formula.
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Volume = πr²h (cubic units); surface area = 2πrh + 2πr² (square units). Different formulas.
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.
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Curved area is 15π and base area is 9π; k = 24.
A pyramid has base area 30 cm² and perpendicular height 8 cm. Find its volume in cm³.
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V = base area × height / 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 $150\pi$