# Height of a cone - This way?

• May 31st 2009, 09:43 AM
Height of a cone - This way?
If you have the net of cone (excluding a circular base).
So it is a sector of a circle, with the radius given.

Can you work out the would be height of the circle by doing -
exterior sector angle / 360 = x
x times diameter of circle = answer

Just wondering if someone could proof it.. using perhaps a different cyclinder to the one below?

http://www.yummiehost.com/images/44144cone.png

(dont need question answering, but thats where the idea comes from)

Thanks,
Alex.

• May 31st 2009, 10:05 AM
earboth
Quote:

If you have the net of cone (excluding a circular base).
So it is a sector of a circle, with the radius given.

Can you work out the would be height of the circle by doing -
exterior sector angle / 360 = x
x times diameter of circle = answer

...

1. You know the radius R of the sector and the central angle $\mu$

2. The arc of the sector is as long as the circumference of the base circle. The base circle has the radius r.

$\dfrac{\mu}{360^\circ} \cdot 2\pi \cdot R = 2\pi\cdot r~\implies~\boxed{r= \dfrac{\mu}{360^\circ} \cdot R}$
• May 31st 2009, 04:32 PM
aidan
Quote:

If you have the net of cone (excluding a circular base).
So it is a sector of a circle, with the radius given.

Can you work out the would be height of the circle by doing -
exterior sector angle / 360 = x
x times diameter of circle = answer

Just wondering if someone could proof it.. using perhaps a different cyclinder to the one below?

http://www.yummiehost.com/images/44144cone.png

(dont need question answering, but thats where the idea comes from)

Alex.

The information given involves a 2nd degree equation. I think that is is not possible to use your suggestion to provide a solution, or a proof.

You are given the radius [call it R] and the interior angle, thus you can compute the arc lengh AB.
The arc length AB becomes the circumference of the base of the cone, then you can compute the radius of the base of the cone [call it r].

You already know the slant distance (the distance from a point on the circumference of the cone to the apex) of the cone.

The height of the cone = $\sqrt{R^2 - r^2}$