Question

Help with finding the height of a cone! In the following diagram, the central angle is measured in radians. Express the height (h) of the cone in terms of theta and R. I know that r = R * theta / 2*pi. Am I right in saying h = 3V / (R^2 * theta / 2) ? Thanks!

Answer 1

\[r = R \theta / 2\]
\[Area of sector: R^2\theta / 2\]
\[h = 3V / (R^2\theta / 2)\]

Answer 2

Would that be correct?

Answer 3

Oops, forgot to square 'r' earlier and so deleting that reply.

\[
R\theta = 2\pi r \\
r = \frac{R\theta}{2\pi} \\
L = R \\
h^2 + r^2 = L^2 \\
h^2 = L^2 - r^2 = R^2 - \left(\frac{R\theta}{2\pi}\right)^2 =
R^2\left(1 - \frac{\theta^2}{4\pi^2}\right) =
R^2\left(\frac{4\pi^2 - \theta^2}{4\pi^2}\right) \\

h = \frac{R}{2\pi}\sqrt{4\pi^2 - \theta^2}
\]