In this chapter we will introduce the concept of cone and show the formula to calculate volume and surface area of cone.

## What is cone ?

It’s a three dimensional figure formed when** infinite line segment from circular base intersect at common point called vertex**.

The shape of the cone is given below;

The above cone has base circle of radius r and all the points from the circle join the point A through straight line.

**Properties of cone**

(a) cone has one circular base

(b) It has one vertex

(c) The vertex A is just above the center of base circle.

(d) Ice-cream cone is one common example of cone in practical life.

**Parts of cone**

The shape of cone has three important elements;

(a)** Radius of base (r)**

It is the radius of circular base of the cone

(b) **Height of cone (h)**

The distance from center of base to apex point (A) is the height of cone

(c) **Slant height (l)**

The distance from apex to any point on circumference of base circle is called slant height.

I hope you understood the basic concept, let us now understand the formulas.

### Volume of cone

The total space captured by the figure is called volume of cone.

**Volume** = \mathtt{\frac{1}{3} \pi r^{2} h}

We get the volume in cubic units ( \mathtt{cm^{3} ,\ m^{3} ,\ etc.} )

### Curved surface area of cone

The area of the slant surface is called curved surface area of cone. It is also called lateral surface area.

**Curved surface area = 𝜋. r. l**

Curved surface area is expressed in square units ( \mathtt{cm^{2} ,\ m^{2} ,\ etc} )

### Total surface area

Cone is made of lateral surface and circular base. So, total surface area is given as;

Total surface area = circular base + Lateral surface**Total surface area = 𝜋.r.(l + r)**

The value of total surface is expressed in square units ( \mathtt{cm^{2} ,\ m^{2} ,\ etc} )

## Questions on volume and surface area of cone

**Question 01**

Find the volume of cone if radius of base is 14 cm and height is 30 cm.

**Solution**

Radius (r) = 14 cm

Height (h) = 30 cm

Volume of cube = \mathtt{\frac{1}{3} \pi r^{2} h}

Putting the values in formula;

\mathtt{Volume\ =\ \frac{1}{3} \times \frac{22}{7} \times 14\times 14\times 30\ }\\\ \\ \mathtt{Volume\ =\ 6160\ cm^{3}}

Hence, the volume of cone is 6160 cu. cm

**Question 02**

Consider the cone of radius 6 cm and height 8 cm. Find the curved surface area of cone.

**Solution**

In the given question;

r = 6 cm

h = 8 cm

Consider triangle AOB. Using Pythagoras theorem;

\mathtt{l^{2} =h^{2} +\ r^{2}}\\\ \\ \mathtt{l^{2} =\ 8^{2} +6^{2}}\\\ \\ \mathtt{l^{2} =\ 64\ +\ 36\ =\ 100}\\\ \\ \mathtt{l\ =\ 10\ cm}

So the slant height is 10 cm.

Now **calculating the curved surface area**.

Curved surface = 𝜋.r. l

Curved surface = (22/7) x 6 x 10 = 188.6 sq cm

Hence, the curved surface of cone is 188.6 sq. cm.

**Question 03**

The curved surface area of cone with radius 5 cm is given as 220 sq. cm. Calculate the slant height of the given cone.

**Solution**

Radius (r) = 5 cm

Curved surface area = 220 sq. cm

Curved surface area = 𝜋.r. l

220 = (22/7) . (5) . l

\mathtt{l=\ \frac{220\times 7}{22\times 5} =14\ cm}

Hence, **the slant height of given cone is 14 cm .**

**Question 04**

Find the area of base of the cone if height and slant height is 21 and 28 cm respectively.

**Solution**

Height (h) = 21 cm

Slant height = 28 cm

Taking triangle AOB and applying Pythagoras theorem.

\mathtt{l^{2} =h^{2} +\ r^{2}}\\\ \\ \mathtt{28^{2} =\ 21^{2} +r^{2}}\\\ \\ \mathtt{r^{2} =784-441}\\\ \\ \mathtt{r^{2} \ =\ 343\ cm}

We know that base of cone is circular in nature.

Area = \mathtt{\pi r^{2}}

Putting the values;

\mathtt{Area\ =\frac{22}{7} \times 343\ =\ 1078\ cm^{2}}

Hence, **area of circular base is 1078 sq. cm**