Why is the volume of sphere four third of cylinder's?

[Jones]

I would also like to know why lateral surface of cone is pi r l and also what is the surface area of sphere and it's proof.

[aero]

The volume of a sphere is V = 4/3 pi r^3, that of a cylinder, V = pi r^2 h, that of a cone, V = 1/3 pi r^2 h. With the condition that h = 2r, the volume of the sphere is 2/3 that of the cylinder, and the volume of the cone is 1/3 that of the cylinder, i.e. V_sphere + V_cone = V_cylinder.

The lateral surface area of a cone can be thought of as 1/2 the area of the equivalent cylinder. The 1/2 comes from taking the average of the circumference at the top and the bottom.

Given the volume of a sphere, we can reason that its surface area is the derivative with respect to radius, since as the radius of the sphere increases, the volume of the sphere increases at a rate proportional to its surface. Applying the power rule from differential calculus immediately gives A = 4 pi r^2.

[Jesse_J]

To compare volumes of a sphere and cylinder, lets look at integrals.

Integrate useing spherical coordinates for the sphere:

dV = r^2 * sin? * dr * d? * dphi
V = r^3/3 [r = 0 to r] * -cos? [? = 0 to pi] * phi [phi = 0 to 2pi]
V = r^3/3 * 2 * 2pi
V= 4pi r^3 /3

For a cylinder use cylindrical coordinates

dV = r * dr * d? * dz
V = (r^2/2) * 2pi * H
V = r^2 * pi * H
and if H = r
V = r^3 * pi

If you don't understand integrals fully it's not possible for me to explain why the cylinder is bigger except that it takes up more space.

As for the surface area of the sphere it is
S = 4*pi*r^2

The proof is the same as the integral i did above except you don't integrate over r because r stays at a constant value.