# The volume of an oblique cone with equal diameter and height `x` is `18pi` cubic cm. Find the height and radius (to the nearest cm).

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The volume of a cone, oblique or right circular is computed using the formula

V = 1/3 x area of base x height

(this can be proved using *calculus -* a cone is a *solid of revolution*, or *Cavalieri's principle*)

Since the base is a circle in this case we have

`V = 1/3 pi r^2 h`

where `r` is the radius of the base and `h` is the height of the cone *perpendicular to the base.* An oblique cone is one where the apex is not perpendicularly above the centre of the base.

Since the diameter `2r` and height `h` are equal, then we have that the radius `r =x/2` and the height `h=x` .

Therefore the volume of the cone is given by

`V = 1/3 pi (x/2)^2 x = 1/3 pi x^3/4 = 1/12 pi x^3`

Now, we are given that `V = 18pi` cubic cm and we wish to find the height`x` and the radius `x/2` . Using the equation for the volume given above we have then that

`1/12 pix^3 = 18pi`

Cancel the `pi`'s out on both sides, giving

`1/12x^3 = 18`

This implies that `x^3 = 12 times 18 = 216` (10 x 18 + 2 x 18 = 180 + 36 = 180 + 20 + 16) which further implies that `x = root(3)(216)`

If it is a 'non-calculator' question, you may know off-by-heart, or have been told in the question that the cube root of 216 is 6.

We have then that `x = 6` cm to the nearest cm so that

height = 6 cm (to the nearest cm)

and radius = 3 cm (to the nearest cm)

(the 'to the nearest cm' part is something of a misleading statement as x is a whole number anyway. If it is a non-calculator question though, one might expect the answer to be a 'nice' whole number anyway).

**height = 6cm ( to the nearest cm)**

**and radius = 3cm (to the nearest cm)**

The formula for the volume of a cone is:

`volume = (1/3)pir^2h`

We know that the volume of this cone is equal to`18picm^2`

and that the height and diameter are equal. Keep in mind that the radius is also equal to half the diameter.

So, if `d=diameter`

`r=d/2`

`h=d`

We can write the volume equation in terms of "d" and solve:

`(1/3)pi(d/2)^2d = 18pi`

Divide by pi and combine like terms:

`(d^3)/12 = 18`

Isolate d: `d^3 = 216`

Cube root:`d=6`

Height is equal to diameter and radius is half of diameter:

Height = 6cm

Radius = 3cm