The volume of a cylinder and a cone

Measures and solids, grade 6 Lesson 13 of 16Learning

The volume of a cylinder and a cone

The volume of a cylinder

$\displaystyle V$ – the volume
$\displaystyle r$ – the base’s radius
$\displaystyle h$ – the height

The volume of a cylinder, like that of a box, is the base’s area times the height.

Fill a cylinder with water, layer after layer, each layer one centimetre high.
Each layer is a disc of water 1 cm thick: its volume is the base’s area, $\displaystyle \pi r^2$ , times 1.
A cylinder $\displaystyle h$ cm high holds $\displaystyle h$ such layers.

\displaystyle V

\displaystyle =

\displaystyle \pi r^2 h

To calculate: $\displaystyle 3.14\times r\times r\times h$ .

Finding a cylinder’s volume

A tin can is a cylinder: its radius is 4 cm, its height 10 cm.
How many cm³ does it hold?

The base’s area: $\displaystyle {3.14\times4\times4}={50.24}$ cm².
The volume: $\displaystyle {50.24\times10}={502.4}$ cm³ – about half a litre.

Exercise 1

Find the cylinder’s volume, in cm³ ( $\displaystyle \pi\approx3.14$ ).

radius 5 cm, height 10 cm

/ – 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

radius 2 cm, height 10 cm

/ – 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

diameter 10 cm, height 4 cm

/ – 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 2

A pot’s radius is 10 cm, and its height 20 cm.
How many litres of water does it hold? ( $\displaystyle \pi\approx3.14$ )

/ – 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

The volume of a cone

Take a hollow cone and a hollow cylinder with the same base and the same height. Fill the cone with sand and pour it into the cylinder: after three times the cylinder is full.

So the volume of a cone is a third of the cylinder’s – the base’s area times the height, divided by 3:

\displaystyle V

\displaystyle =

\displaystyle \frac{\pi r^2 h}{3}

Finding a cone’s volume

An ice-cream cone has a radius of 3 cm and a height of 10 cm.
How many cm³ does it hold?

A cylinder with the same base and height: $\displaystyle {3.14\times3\times3\times10}={282.6}$ cm³.
The cone is a third of it: $\displaystyle {282.6:3}={94.2}$ cm³.

Exercise 3

Find the cone’s volume, in cm³ ( $\displaystyle \pi\approx3.14$ ).

radius 6 cm, height 10 cm

/ – 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

radius 3 cm, height 5 cm

/ – 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 4

A cylinder and a cone have the same base and the same height.
The cone’s volume is 50 cm³.
What is the cylinder’s volume?

/ – 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 5

A cylinder’s volume is 600 cm³, and its height is 12 cm.
What is the area of its base?

/ – 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

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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
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{2}{3}\times\pi r^2\times2r$