The cylinder

Circle and solids, grade 8 Lesson 3 of 7Learning

The cylinder

A cylinder is a solid like a drink can, a pipe or a round candle.

It has two bases: two equal discs, parallel, one above the other.
Between the bases is the curved surface, rolled around them.
$\displaystyle r$ – the radius of the base.
$\displaystyle h$ – the height: the distance between the two bases.

The cylinder’s net

We cut the curved surface along a straight line from base to base, and spread it out on the table – it is a rectangle. Together with the two bases this is the cylinder’s net:

One side of the rectangle is the height, $\displaystyle h$ .
The other side went exactly once around the base, so it is as long as the base’s circumference, $\displaystyle 2\pi r$ .

Curved surface area and surface area

$\displaystyle L$ – the curved surface area
$\displaystyle A$ – the surface area

The curved surface area is the rectangle’s area – its length times its width:

\displaystyle L

\displaystyle =

\displaystyle 2\pi r h

The cylinder’s surface area is the area of the whole net – the curved surface and the two bases:

\displaystyle A

\displaystyle =

\displaystyle 2\pi r^2+2\pi r h

Finding the surface area

A can’s radius is 4 cm, and its height is 10 cm.
How much tin does it take to make it? ( $\displaystyle \pi\approx3.14$ )

The curved surface: $\displaystyle {2\times3.14\times4\times10}={251.2}$ cm².
The two bases: $\displaystyle {2\times3.14\times4^2}={100.48}$ cm².
The surface area: $\displaystyle {251.2+100.48}={351.68}$ cm².

Exercise 1

Find, in cm² ( $\displaystyle \pi\approx3.14$ ):

The label around a can of radius 5 cm and height 12 cm (the curved surface alone)

/ – 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

The surface area of a closed cylinder of radius 2 cm and 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

The volume of a cylinder

$\displaystyle V$ – the volume
1 litre = 1000 cm³

A box’s volume is its length times its width times its height – the area of its base times its height. So is a cylinder’s:

Think of the cylinder as a stack of thin discs, each 1 cm thick.
The volume of each layer is the base area times 1, and there are $\displaystyle h$ layers.
So the volume is the base area, $\displaystyle \pi r^2$ , times the height:

\displaystyle V

\displaystyle =

\displaystyle \pi r^2 h

Finding the volume

How much drink fits in the same can – radius 4 cm and height 10 cm?

The base area: $\displaystyle {3.14\times4^2}={50.24}$ cm².
The volume: $\displaystyle {50.24\times10}={502.4}$ cm³ – about half a litre.

Exercise 2

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

/ – 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

Exercise 3

A glass’s radius is 3 cm.
$\displaystyle 282.6$ cm³ of juice are poured into it.
How high does the juice reach? Write an equation and solve it ( $\displaystyle \pi\approx3.14$ ).

Exercise 4

Two containers in the shape of a cylinder:

container A: radius 5 cm and height 10 cm;
container B: radius 10 cm and height 5 cm.

Which container holds more? How many times as much?

/ – 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: circle and solids, grade 8

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The circle and the disc

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
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$

The cylinder

The curved surface area – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle L$ $\displaystyle =$ $\displaystyle 2\pi r h$
The surface area – the two bases and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2\pi r^2+2\pi r h$
The volume – the base area times the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2 h$

The cone

The slant height $\displaystyle l$ – the hypotenuse of a triangle whose legs are the radius $\displaystyle r$ and the height $\displaystyle h$
$\displaystyle l^2$ $\displaystyle =$ $\displaystyle r^2+h^2$
The curved surface area
$\displaystyle L$ $\displaystyle =$ $\displaystyle \pi r l$
The surface area – the base and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2+\pi r l$
The volume – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2 h}{3}$

Pythagoras' theorem in space

The diagonal of a box $\displaystyle d$ – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three edges
$\displaystyle d^2$ $\displaystyle =$ $\displaystyle a^2+b^2+c^2$
In a square pyramid – $\displaystyle k$ the height of a side face, $\displaystyle h$ the pyramid’s height, $\displaystyle a$ the side of the base
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$