Question

A cone and a cylinder have the same height and their bases are congruent circles

If the volume of the cylinder is 90 cm^3,
what is the volume of the cone?

Answer 1

volume of the cone = 30 cm³

Explanation:

first write both the formula for finding the volume a cone and a cylinder

volume of a cone =
`(1)/(3)` πr² h

where r is the radius of the cone and h is the height of the cone

volume of a cylinder = πr²h

where r is the radius of the cylinder and h is the height of a cylinder

since the cone and the cylinder has the same height and they are congruent which implies they have the same radius, then

volume of a cone =
`(1)/(3)` πr² h

=
`(1)/(3)` (volume of a cylinder)

from the question given, volume of the cylinder = 90 cm³

volume of a cone =
`(1)/(3)` (volume of a cylinder)

=
`(1)/(3)` × 90

= 30 cm³

Therefore, volume of the cone = 30 cm³

Answer 2

`30cm^3`

Explanation:

the volume of a cylinder is given by:

`v_(cylinder)=\pi r^2 h`

and the volume of a cone is given by:

`v_(cone)=(\pi r^2 h)/(3)`

since both have the same height and radius, we can solve each equation for
`r^2h` (because this quantity is the same in both figures) and then match the expressions we find:

from the cylinder's volume formula:

`r^2h=(v_(cylinder))/(\pi)`

and from the cone's volume formula:

`r^2h=(3 v_(cone))/(\pi)`

matching the two previous expressions:

`(v_(cylinder))/(\pi) =(3v_(cone))/(\pi)`

we solve for the volume of a cone
`v_(cone)`:

`v_(cone)=(\pi v_(cylinder))/(3\pi) \\\\v_(cone)=(v_(cylinder))/(3)`

substituting the value of the cylinder's volume
`v_(cylinder)=90cm^3`

`v_(cone)=(90cm^3)/(3) \\\\v_(cone)=30cm^3`