Question

Find the volume of the figure: a cone with a square pyramid of the same height cut out. The pyramid has height l, and its square base has area l2.

Answer 1

The volume of the figure is
`(l^(3))/(3)[(1)/(2)\pi-1]\ units^(3)`

Explanation:

we know that

The volume of the figure is equal to the volume of the cone minus the volume of the square pyramid

step 1

Find the volume of the cone

The volume of the cone is equal to

`V=(1)/(3)\pi r^(2) h`

we have

`r=l(√(2))/(2)\ units`

`h=l\ units`

substitute

`V=(1)/(3)\pi (l(√(2))/(2))^(2)l`

`V=(1)/(3)\pi ((l^(3))/(2) )`

`V=(1)/(6)\pi (l^(3))\ units^(3)`

step 2

Find the volume of the square pyramid

`V=(1)/(3)Bh`

we have

`B=l^(2)\ units^(2)`

`h=l\ units`

substitute

`V=(1)/(3)(l^(2))l`

`V=(1)/(3)(l^(3))\ units^(3)`

step 3

Find the difference

`(1)/(6)\pi (l^(3))-(1)/(3)(l^(3))=(l^(3))/(3)[(1)/(2)\pi-1]\ units^(3)`