54
Volume of a rectangular parallelepiped and a cube
Textbook: Part 2, lesson 54, pp. 66–70
GoalThe idea of volume; the volume of a rectangular parallelepiped and a cube; units of volume.
New words
volume · hajmunit cube · birlik kubcubic centimetre (cm³) · santimetr kub (cm³)area of the base · asos yuzi
Explanation
The volume of a body shows how many unit cubes fit into it. A cube with edge 1 cm has volume 1 cm³. The volume of a rectangular parallelepiped is the product of its three dimensions: V = a · b · c. It can also be written V = S · H, where S is the area of the base and H is the height. For a cube V = a³. Among the units of volume, 1 dm³ = 1000 cm³ and 1 m³ = 1000 dm³. For liquids, 1 dm³ is taken to be 1 l. When calculating a volume, all dimensions must be in the same unit.
Worked examples
A box 5 cm by 4 cm by 3 cm: the bottom layer holds 5 · 4 = 20 cubes, with 3 layers: 20 · 3 = 60. V = 5 · 4 · 3 = 60 cm³.
V = 336 cm³ and base area S = 42 cm². Height H = V : S = 336 : 42 = 8 cm. Cube with a = 4 dm: V = 4 · 4 · 4 = 64 dm³ = 64 l.
Class activity
“Layer by layer”: build different boxes from 12 identical small cubes (for example 3 · 2 · 2). Write down the dimensions of each and check V = a · b · c.
Practice
1
Volume (cm³) of a parallelepiped with dimensions 12 cm, 5 cm and 3 cm?
180
2
V = 900 cm³ and the base area is 75 cm². What is the height (cm)?
12
3
How many dm³ is 3 m³?
3000
4
Why must dimensions be converted to the same unit before calculating volume?
Otherwise the product is in an incorrect unit: for example, m and cm cannot be mixed.