Questions on surface area of cone

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