Question

Derive the formula for volume of cone and sphere. (Class 9)

Answer 1

Formula for cone=1/3*π*radius*radius*height
formula for sphere=4/3*π*radius*radius*radius

Answer 2

                                               SPHERE

Suppose two regions in 3-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross -section of equal area, then the two regions have equal volumes.

Okay, so suppose we have hemisphere of radius R. Suppose also that we have a cylinder of height and radius R. Finally suppose we cut a cone of height and radius R from the cylinder and call the resulting shape T..

image 1

Consider a cross-section of T at height h. The result is a “washer” with inner radius h and full radius of say, r. Hence we can compute its area by \pi r^2 - \pi h^2 = \pi (r^2-h^2).Then if we consider the cross-section at the same height we have a radius of r'. However by looking at the following picture,

image 2

We see that in fact the Pythagorean Theorem tells us r'^2 = r^2 - h^2. Therefore the area is \pi r'^2 = \pi (r^2 - h^2). By applying Cavalieri’s Principle we see that the volume of a hemisphere is the volume of cylinder minus the volume of a cone. Or V = \pi r^3 - 1/3 \pi r^3 = 2/3 \pi r^3. Finally, we conclude the volume of a full sphere is V = 4/3 \pi r^3

                                             CONE

consider a cylinder of unit volume $\pi$ * r² * h

if we carefully look a cone and hemisphere have same base and made a cylinder when combine .

NOTE - YOU HAVE TO IMAGINE THAT THEV HEMISPHERE IS GOING INSIDE THE CONE.

now we know that volume of hemisphere = 2/3 * $\pi$ * r³

= 2/3 * $\pi$ * r² * h [because r = h in hemisphere]

hence volume of cone = $\pi$ * r² * h - 2/3 * $\pi$ * r² * h

= 1/3 * $\pi$ * r² * h

hope it helped you