Surface Areas and Volumes
Volume of combined solids
Find the volume of a solid made of two or more shapes.
Volume is how much space something fills. When solids are joined, volumes simply add: unlike surface area, nothing disappears.
| Solid | Volume |
|---|---|
| cuboid | l × b × h |
| cylinder | πr²h |
| cone | ⅓ πr²h (a third of the cylinder on the same base and height) |
| sphere | 4/3 πr³ |
| hemisphere | ⅔ πr³ |
Example: Shanta's shed is a cuboid (15 × 7 × 8 m) with a half cylinder on top (diameter 7 m, length 15 m).
- Air in the empty shed = cuboid + half cylinder = 1128.75 m³.
- The machines take up 300 m³ and 20 workers take about 0.08 m³ each (1.6 m³).
- So the air left = 1128.75 – 301.6 = 827.15 m³.
Things that take up room must be subtracted. A juice seller's glass is 5 cm wide and 10 cm tall, but it has a raised hemisphere in its base.
- It seems to hold πr²h = 196.25 cm³.
- It really holds 196.25 – ⅔πr³ = 163.54 cm³. The customer gets less than it looks.
Dropping things into water: a solid sunk in a full vessel pushes out its own volume of water. To find how many lead shots spill a quarter of a cone of water, divide (the water spilled) by (one shot's volume).
Mass = volume × mass of 1 cm³. If 1 cm³ of iron weighs 8 g, a pole of 111 532.8 cm³ weighs 892 262.4 g, about 892 kg.
Watch the units: area is in cm² (two lengths multiplied), volume in cm³ (three). Doubling the radius of a sphere makes its surface area 4 times bigger and its volume 8 times bigger.
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