In this post we will try to solve questions related to surface area cone.

**Lateral surface area of cone**

The** area of the curved part** of the cone is called lateral surface area.

**Lateral surface area = 𝜋.r.l**

Where;

r = radius of base

l = slant height of cone

**Total surface area of cone**

The total surface area includes the **area of the base and curved part of the cone**.

**TSA = 𝜋. r ( l + r)**

You need to memorize both the formulas as they would help to solve below questions.

## Surface area of cone – Solved examples

**Question 01**

Find the curved surface area of cone if radius is 7 cm and slant height is 10 cm.

**Solution**

Radius (r) = 7 cm

Slant height (l) = 10 cm

Area of curved surface = 𝜋. r. l

Putting the values in formula;

Area of curved surface = \mathtt{\frac{22}{7} \times 7\times 10\ =220\ cm^{2}}

Hence, **220 sq. cm is the area of curved surface**.

**Question 02**

Consider the cone with radius 3 cm and height 4 cm. Find the curved surface area of the cone.

**Solution**

Radius (r) = 3 cm

Height (h) = 4 cm

To find the curved surface area, we have to first find the slant height.

Applying Pythagoras theorem in triangle OAB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{l^{2} =4^{2} +3^{2}}\\\ \\ \mathtt{l^{2} =\ 16\ +\ 9=\ 25}\\\ \\ \mathtt{l\ =\ 5\ cm\ }

So, the **slant height (l) is 5 cm**.

Let’s now calculate the lateral surface area.**Area of curved surface = 𝜋. r. l **

Putting the values;

Area = \mathtt{\frac{22}{7} \times 3\times 5\ =47.14\ cm^{2}}

Hence, **47.14 is the curved surface of give cone**.

**Question 03**

The height of the cone is given as 8 cm. If the slant height is 10 cm then find the area of base of the cone.

**Solution**

Heigh (h) = 8 cm

Slant height (l) = 10 cm

Let’s first calculate the radius of base of cone.

Applying Pythagoras theorem on triangle OAB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{10^{2} =8^{2} +r^{2}}\\\ \\ \mathtt{r^{2} =100-64=36\ }\\\ \\ \mathtt{r\ =\ 6\ cm}

So, the radius of circular base is 6 cm.

We know that cone is a circular base and its area is calculated by following formula;

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

Area = \mathtt{\frac{22}{7} \times 36\ =\ 113.14}

Hence, **113.14 sq. cm is the area of base of cone**.

**Question 04**

Find the total surface area of cone with radius 5 cm and height 12 cm.

**Solution**

Radius (r) = 5 cm

Height (h) = 12 cm

We have to first calculate the slant height.

Applying Pythagoras theorem in triangle AOB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{l^{2} =12^{2} +5^{2}}\\\ \\ \mathtt{l^{2} =144+25=169}\\\ \\ \mathtt{l\ =\ 13\ cm\ }

So the **slant height is 13 cm**.

The formula for total surface area is given as;**TSA = 𝜋. r ( l + r) **

\mathtt{T.S.A=\ \frac{22}{7} \times 5\times ( 13+5) \ }\\\ \\ \mathtt{T.S.A\ =\ \frac{22}{7} \times 5\times 18}\\\ \\ \mathtt{T.S.A\ =\ 282.9\ cm^{2}}

Hence, **282.9 sq cm is the total surface area of the cone.**

**Question 05**

Consider a cone with diameter 70 cm and lateral surface area 4070 sq cm. Find the slant height of the cone.

**Solution**

Diameter = 70 cm

Radius(r) = 70 / 2 = 35 cm

Lateral surface area = 4070 sq cm

The formula for lateral surface is given as;**Lateral surface area = 𝜋. r . l **

4070 = (22 / 7) . 35 . l

l = \mathtt{\frac{4070\times 7}{22\times 35} =37\ cm}

Hence, **37 cm is the slant height of the given cone**.

**Question 06**

A tent is in the form of cone of radius 3 meter and height 4 meter. If the cost of canvas is 7$ per square meter then calculate the cost of making the tent.

**Solution**

In conical tent, the canvas is used to make the curved surface. So, to calculate the cost, we have to find the curved surface area.

Let’s calculate the slant height first;

Applying Pythagoras theorem in triangle AOB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{l^{2} =4^{2} +3^{2}}\\\ \\ \mathtt{l^{2} =16+9=25}\\\ \\ \mathtt{l\ =\ 5\ m\ }

So the **slant height of the tent is 5 meter**.

Calculating the** curved surface area of tent**.

Curved surface area = 𝜋. r. l

Curved surface area = \mathtt{\frac{22}{7} \times 3\times 5\ =\frac{330}{7}}

Hence, **330/7 sq meter is the curved surface area.**

Calculating the** cost of canvas**

1 sq meter cost ⟹ 7 $

330/7 sq meter cost ⟹ 330/7 x 7 = 330 $

Hence, the **total cost of making tent is 330$ **