Question

A cone has 15 cm and 25 cm as their radius and slant height. Find its lateral surface area and total surface area. $\left( {\pi = 3.14} \right)$

Answer 1

Hint: We are given the radius and the slant height of a cone. Then we can find the lateral surface area using the equation $LSA = \pi rl$. Then we can find the area of the base using the radius of the cone. Then we can find the total surface area by taking the sum of the lateral surface area and area of the base.

Complete step by step solution:
We can represent the cone by drawing a diagram.

We are given that radius is 15 cm.
 $ \Rightarrow r = 15$ .
It is also given that the slant height of the cone is 25cm.
 $ \Rightarrow l = 25$ .
We know that lateral surface area of the cone is given by the equation,
 $LSA = \pi rl$ .
Now we can substitute the values of r, l and $\pi = 3.14$ as given in the question.
 $ \Rightarrow LSA = 3.14 \times 15 \times 25$
After doing the multiplication, we get,
 $ \Rightarrow LSA = 1177.5c{m^2}$
Hence the lateral surface area is $1177.5c{m^2}$ .
Now we can find the area of the base. We know that the base of a cone is a circle with radius r.
Then its area is given by the equation,
 $A = \pi {r^2}$
Now we can substitute the values of r and $\pi = 3.14$ as given in the question.
 $ \Rightarrow A = 3.14 \times {15^2}$
After doing the multiplication, we get,
 $ \Rightarrow A = 706.5c{m^2}$
Thus, the area of the base is $706.5c{m^2}$ .
Now the total surface area is given by the sum of the lateral surface area and base area. So, the total surface area will become,
 $ \Rightarrow TSA = LSA + A$
On substituting the values, we get,
 $ \Rightarrow TSA = 1177.5 + 706.5$
 $ \Rightarrow TSA = 1884c{m^2}$
So, the total surface area is $1884c{m^2}$

Therefore, the lateral surface area and total surface area of the given cone are $1177.5c{m^2}$ and $1884c{m^2}$ respectively.

Note: Alternate method to find the total surface area of the cone is by using the direct equation.
We know that lateral surface area of the cone is given by the equation,
 $TSA = \pi r\left( {l + r} \right)$ .
Now we can substitute the values of r, l and $\pi = 3.14$ as given in the question.
 \[ \Rightarrow TSA = 3.14 \times 15 \times \left( {25 + 15} \right)\]
 \[ \Rightarrow TSA = 3.14 \times 15 \times 40\]
After doing the multiplication, we get,
 $ \Rightarrow TSA = 1884c{m^2}$
Hence the lateral surface area is $1884c{m^2}$ .