Question

How is the formula for the volume of a sphere derived?

Answer 1

Answer:

The sphere and the cylinder have EQUAL heights.

A right triangle is created by the radius of the sphere r, the radius of the cross section y, and the distance between the center of the sphere and the center of the cross section x.

So  y2=r2−x2 by the Pythagorean theorem.  

2 - The area of the cross section of the sphere, and every cross section parallel to it, is $\pi r^2 - \pi x^2$.

Each cross section of the cylinder with two cones removed is the shape of an annulus with an area of  πr2−πx2 .

By Cavalieri's principle, this sphere and this cylinder with two cones removed have equal volumes.  

The volume of the cylinder with radius r and height 2r is 2πr3 .  

3 - The volume of each cone with radius r and height r is $\frac{1}{3}\pi r^3$.

The volume of the cylinder with two cones removed is 43πr3 .

Therefore, the volume of the sphere is 43πr3 .

Answer 2

Answer:

Step-by-step explanation:

From the picture we find that the sphere and the cylinder has equal heights.

i.e. cylinder has height = 2r.

x,y and r form a right triangle with hypotenuse as r.

Hence pythagorean theorem is satisfied

Area of the cross section parallel to it is area of circle with radius y

= $\pi y^2\\=\pi (r^2-x^2)\\=\pir^2-\pix^2$

Volume of cylinder when 2 cones removed = $\frac{4\pir^3}{3}$

Hence volume of sphere = $\frac{4\pir^3}{3}$