The volume of a box

Measures and solids, grade 6 Lesson 12 of 16Learning

The volume of a box

Estimating volume

Volume is how much room a solid takes up – or how much fits inside it. It is worth estimating a volume before calculating it:

An ordinary cup holds about 250 cm³ – a quarter of a litre.
A litre bottle holds about 4 cups.
A 330 cm³ drink can holds a cup and a bit.
The unit is chosen by the size of the solid: medicine on a spoon – in ml; a bucket – in litres; a pool – in m³.

Exercise 1

How many 250 cm³ cups does each container hold?

a 1.5-litre bottle

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a 2-litre jug

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a 5-litre pot

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A box of cubes

Shuli built a box of centimetre cubes: 4 cubes long, 3 wide, and 2 layers high.

Each layer has $\displaystyle {4\times3}={12}$ cubes.
There are 2 layers, so the box has $\displaystyle {12\times2}={24}$ cubes.
The box’s volume is 24 cm³.

The volume of a box

$\displaystyle V$ – the volume
$\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ – the length, width and height
$\displaystyle S$ – the base’s area
$\displaystyle h$ – the height

The volume of a box is its length times its width times its height:

\displaystyle V

\displaystyle =

\displaystyle a\times b\times c

The length times the width is the base’s area – the number of cubes in one layer. The height is the number of layers. So the volume is also the base’s area times the height:

\displaystyle V

\displaystyle =

\displaystyle S\times h

Any face of the box can be its base: $\displaystyle 4\times3\times2$ , $\displaystyle 3\times2\times4$ and $\displaystyle 2\times4\times3$ are the same volume, 24.
All the edges (the box’s sides) must be in the same unit: cm give cm³, metres give m³.

Finding the volume

A fish tank is a box 50 cm long, 30 cm wide and 40 cm high.
How many litres of water does it hold?

The volume: $\displaystyle {50\times30\times40}={60{,}000}$ cm³.
A litre is 1000 cm³, so it holds $\displaystyle {60{,}000:1000}={60}$ litres.

Exercise 2

Find the volume.

a box 5 cm by 4 cm by 3 cm; in cm³

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a cube whose edge is 6 cm; in cm³

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a room 4 m long, 3 m wide and 2.5 m high; in m³

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a box: its base 20 cm², its height 8 cm; in cm³

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Different solids, the same volume

Different solids can have the same volume. From 12 centimetre cubes one can build, for example:

a long row, $\displaystyle 12\times1\times1$ ;
a box $\displaystyle 3\times2\times2$ ;
a solid that is not a box at all, like a staircase.

Each of them has a volume of 12 cm³.

Exercise 3

Design three different boxes, each with a volume of 24 cm³.
The edges are whole numbers of cm.

Changing the edges

When one edge of a box is doubled, the volume doubles. When all three edges are doubled, the volume is 8 times as big:

A box $\displaystyle 2\times3\times4$ : volume 24.
Only the length doubled, $\displaystyle 4\times3\times4$ : volume 48 – twice as much.
All the edges doubled, $\displaystyle 4\times6\times8$ : volume 192 – $\displaystyle {2\times2\times2}={8}$ times as much.

Exercise 4

One cube’s edge is 3 cm, another’s 6 cm.
How many times the first cube’s volume is the second’s?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Volume and surface area

The surface area of a solid is the sum of the areas of all its faces – how much paper it takes to wrap it. Solids with the same volume can have different surface areas:

Four cubes in a row: 4 squares on top, 4 underneath, 4 in front, 4 at the back and 2 at the ends – a surface area of 18 cm².
Four cubes in a $\displaystyle 2\times2$ square: 4 on top, 4 underneath and 2 on each of the four sides – 16 cm².
Both have a volume of 4 cm³.
The more compact the solid – the closer to a cube – the smaller its surface area.

Exercise 5

Solids are built from 8 centimetre cubes.
What is each solid’s surface area, in cm²?

a row of 8 cubes

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

a big cube,

\displaystyle 2\times2\times2

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 6

A pool is a box 6 m long and 4 m wide.
It holds 36 m³ of water.
How deep is the pool?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: measures and solids, grade 6

Open as a page

The circle and the disc

The diameter $\displaystyle d$ is two radii $\displaystyle r$
$\displaystyle d$ $\displaystyle =$ $\displaystyle 2\times r$
The number pi – the value to calculate with
$\displaystyle \pi\approx3.14$
The circumference of a circle – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference of a circle – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$

Volumes

The volume of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\times b\times c$
The volume of a box, a prism or a cylinder – $\displaystyle S$ the area of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\times h$
The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2 h$
The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\times h}{3}$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2 h}{3}$
The volume of a sphere – two thirds of the cylinder around it
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{2}{3}\times\pi r^2\times2r$