Question

How can we derive Total Surface Area and volume of a cone and sphere?

Answer 1

Surface Area of a Right Circular Cone
A cone with height , slant height and base radius is given below.

Following figure shows how an opened sheet of paper which is used to make a cone looks.

We know that, area of each triangle $=\frac{1}{2}\times$ Base of each triangle $\times$ height
Total area of the sheet, which is the lateral surface area of the cone,
$=\frac{1}{2}\times l(b_1+b_2+...)$
Since base of the cone is a circle with radius r,
$b_1+b_2+b_3...=$ circumference of the base circle $=2\pi r$
Therefore, lateral surface area $=\frac{1}{2}\times 2\pi r\times l=\pi rl$
Now, if the base of the cone is closed. Then,
Total surface area of the cone = Lateral surface area + Area of the base
$=\pi rl+\pi r^2=\pi r(l+r)$
If a cone and a cylinder have same base radius and height, then;
Volume of cone $=\frac{1}{3}\times$ (Volume of cylinder) $=\frac{1}{3}\pi r^{2}h$
Volume of a cone with radius r and height h is $=\frac{1}{3}\pi r^{2}h$