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Question

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


Solution

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 =12× Base of each triangle × height
Total area of the sheet, which is the lateral surface area of the cone,
=12×l(b1+b2+...)
Since base of the cone is a circle with radius r,
b1+b2+b3...= circumference of the base circle =2πr
Therefore, lateral surface area =12×2πr×l=πrl
Now, if the base of the cone is closed. Then,
Total surface area of the cone = Lateral surface area + Area of the base
=πrl+πr2=πr(l+r)
If a cone and a cylinder have same base radius and height, then;
Volume of cone =13× (Volume of cylinder) =13πr2h
Volume of a cone with radius r and height h is =13πr2h

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