Volume of cuboid, cube and cylinder

Sur­face area tells us how much mate­r­ial cov­ers a solid. Vol­ume tells us how much space it takes up, and capac­ity how much a hol­low con­tainer can hold. Water tanks, grain bins, pipes and brick walls are all vol­ume prob­lems. This les­son gives the vol­umes of cuboids, cubes and cylin­ders, explains the units, and uses them in prac­ti­cal ques­tions.

Units of vol­ume and capac­ity#

Vol­ume is mea­sured in cubic units: a cube of edge $1$ cm has vol­ume $1$ cm³.

$$1 \text{ m}^3 = 100 \times 100 \times 100 \text{ cm}^3 = 1000000 \text{ cm}^3$$

Capac­ity is usu­ally given in litres:

$$1 \text{ litre} = 1000 \text{ cm}^3, \qquad 1 \text{ m}^3 = 1000 \text{ litres}$$

Cuboid and cube#

$$\text{Volume of cuboid} = l \times b \times h, \qquad \text{Volume of cube} = a^3$$

Both come from the same idea: vol­ume $=$ area of the base $\times$ height.

Exam­ple 1#

A water tank is $6$ m long, $4$ m wide and $2.5$ m deep. How many litres can it hold?

$$\text{Volume} = 6 \times 4 \times 2.5 = 60 \text{ m}^3 = 60 \times 1000 = 60000 \text{ litres}$$

Exam­ple 2#

A cube has vol­ume $343$ cm³. Find its edge and its total sur­face area.

$a^3 = 343$, so $a = 7$ cm. Total sur­face area $= 6 \times 7^2 = 294$ cm².

Exam­ple 3#

A wall $15$ m long, $0.3$ m thick and $4$ m high is built of bricks mea­sur­ing $25$ cm by $15$ cm by $8$ cm. How many bricks are needed (ignor­ing mor­tar)?

Work in metres. The brick is $0.25$ m by $0.15$ m by $0.08$ m.

$$\text{Volume of wall} = 15 \times 0.3 \times 4 = 18 \text{ m}^3$$

$$\text{Volume of one brick} = 0.25 \times 0.15 \times 0.08 = 0.003 \text{ m}^3$$

$$\displaystyle \text{Number of bricks} = \frac{18}{0.003} = 6000$$

Always change all lengths to the same unit before mul­ti­ply­ing.

Cylin­der#

A cylin­der also has vol­ume $=$ base area $\times$ height, and its base is a cir­cle:

$$\text{Volume of cylinder} = \pi r^2 h$$

Exam­ple 4#

Find the vol­ume of a cylin­der of radius $7$ cm and height $10$ cm, and its capac­ity in litres.

$$\displaystyle V = \frac{22}{7} \times 7 \times 7 \times 10 = 1540 \text{ cm}^3 = \frac{1540}{1000} = 1.54 \text{ litres}$$

Hol­low cylin­ders#

A pipe is a hol­low cylin­der with outer radius $R$ and inner radius $r$. The mate­r­ial in it is the big cylin­der minus the hole:

$$\text{Volume of material} = \pi R^2 h - \pi r^2 h = \pi(R^2 - r^2)h$$

Exam­ple 5#

A metal pipe is $70$ cm long, with outer radius $4$ cm and inner radius $3$ cm. Find the vol­ume of metal in it.

$$\displaystyle V = \frac{22}{7} \times (16 - 9) \times 70 = \frac{22}{7} \times 7 \times 70 = 1540 \text{ cm}^3$$

Exam­ple 6#

The vol­ume of a cylin­der is $9240$ cm³ and its height is $15$ cm. Find its radius.

$$\displaystyle \frac{22}{7} \times r^2 \times 15 = 9240 \implies r^2 = \frac{9240 \times 7}{22 \times 15} = 196 \implies r = 14 \text{ cm}$$

Prac­tice#

1. Find the vol­ume of a cuboid $12$ cm by $8$ cm by $5$ cm.
2. Find the vol­ume of a cube of edge $9$ cm.
3. A tank is $4$ m long, $3$ m wide and $2$ m deep. How many litres does it hold?
4. Find the vol­ume of a cylin­der of radius $14$ cm and height $15$ cm.
5. A cylin­der has vol­ume $1540$ cm³ and height $10$ cm. Find its radius.
6. A hol­low cylin­der is $14$ cm long, with outer radius $10$ cm and inner radius $8$ cm. Find the vol­ume of mate­r­ial in it.
7. A wall $12$ m long, $20$ cm thick and $3$ m high is built of bricks $24$ cm by $10$ cm by $8$ cm. How many bricks are needed?
8. A cylin­dri­cal tank has diam­e­ter $2.8$ m and height $5$ m. Find its capac­ity in litres.

Answers#