The cone

Circle and solids, grade 8 Lesson 4 of 7Learning

The cone

A cone is a solid like an ice-cream cone, a party hat or a traffic cone.

It has one base – a disc. $\displaystyle r$ is its radius.
The apex is the point at the top of the cone.
$\displaystyle h$ – the height: the segment from the apex to the centre of the base. It is perpendicular to the base.
$\displaystyle l$ – the slant height: a segment from the apex to a point on the circle around the base. All of a cone’s slant heights are equal.

The height, the radius and the slant height

The height, the radius and the slant height are the sides of a right-angled triangle: the right angle is at the centre of the base, since the height is perpendicular to the base.
The slant height is opposite the right angle, so it is the hypotenuse. By Pythagoras’ theorem:

\displaystyle l^2

\displaystyle =

\displaystyle r^2+h^2

For example, $\displaystyle {r}={6}$ cm and $\displaystyle {h}={8}$ cm:

$\displaystyle {l^2}={6^2+8^2}={36+64}={100}$ .
$\displaystyle {l}={\sqrt{100}}={10}$ cm.

Exercise 1

Find, in cm:

The slant height of a cone of radius 5 cm and height 12 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

The height of a cone of slant height 17 cm and radius 8 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 cone

A cone’s volume is a third of the volume of the cylinder with the same base and the same height.

An experiment shows it: fill the cone with water and pour it into the cylinder. After three times the cylinder is exactly full.
The cylinder’s volume is $\displaystyle \pi r^2h$ , so:

\displaystyle V

\displaystyle =

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

Finding the volume

A cone’s radius is 3 cm, and its height is 10 cm.
What is its volume? ( $\displaystyle \pi\approx3.14$ )

The base area: $\displaystyle {3.14\times3^2}={28.26}$ cm².
The volume: $\displaystyle {28.26\times10:3}={94.2}$ cm³.

Exercise 2

An ice-cream cone’s radius is 3 cm, and its height is 12 cm.
How many cm³ of ice cream fit inside it – without the scoop on top? ( $\displaystyle \pi\approx3.14$ )

/ – 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 cone’s radius is 6 cm, and its slant height is 10 cm.
What is its volume? ( $\displaystyle \pi\approx3.14$ )

/ – 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 cone’s net

We cut the cone’s curved surface along a slant height, and spread it out. Together with the base this is the cone’s net:

The curved surface is a sector of a disc of radius $\displaystyle l$ : every point on its edge is as far from the apex as the slant height.
The sector’s arc went once around the base, so its length is the base’s circumference, $\displaystyle 2\pi r$ .

The curved surface area

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

The sector is part of a whole disc of radius $\displaystyle l$ :

The whole disc’s circumference is $\displaystyle 2\pi l$ , and the sector’s arc is $\displaystyle 2\pi r$ . So the sector is the part $\displaystyle {\frac{2\pi r}{2\pi l}}={\frac{r}{l}}$ of the disc.
The whole disc’s area is $\displaystyle \pi l^2$ , so the sector’s area is $\displaystyle {\frac{r}{l}\cdot\pi l^2}={\pi r l}$ .

\displaystyle L

\displaystyle =

\displaystyle \pi r l

The cone’s surface area is the base and the curved surface together:

\displaystyle A

\displaystyle =

\displaystyle \pi r^2+\pi r l

Finding the surface area

A cone’s radius is 6 cm, and its slant height is 10 cm.
What is its surface area? ( $\displaystyle \pi\approx3.14$ )

The curved surface: $\displaystyle {3.14\times6\times10}={188.4}$ cm².
The base: $\displaystyle {3.14\times6^2}={113.04}$ cm².
The surface area: $\displaystyle {188.4+113.04}={301.44}$ cm².

Exercise 4

A party hat is a cone with no base.
Its radius is 10 cm, and its height is 24 cm.
How many cm² of paper does it take to make it? ( $\displaystyle \pi\approx3.14$ )

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Exercise 5

A cone’s radius is 3 cm, and its slant height is 5 cm.
What is its surface area, in cm²? ( $\displaystyle \pi\approx3.14$ )

/ – 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

Formulas: circle and solids, grade 8

Open as a page

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
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$