Question

A cone with labeled dimensions.
Find the length of the height of the cone.

Answer 1

I don't see the diagram — could you upload it or tell me the labeled values?

Meanwhile here are the common ways to get the cone's height depending on what is labeled:

- If you are given the slant height s and the base radius r:
h = sqrt(s^2 − r^2) (Pythagorean theorem on the cross‑sectional right triangle).

- If you are given the volume V and the base radius r:
V = (1/3)πr^2h ⇒ h = 3V / (π r^2).

- If you are given a lateral (side) area A_L and the base radius r:
A_L = π r s ⇒ s = A_L/(π r) and then h = sqrt(s^2 − r^2).

- If you are given an angle θ between the axis and a slant line and the slant s:
h = s cos θ (and r = s sin θ).

Example: if s = 13 and r = 5, then h = sqrt(13^2 − 5^2) = sqrt(169 − 25) = sqrt(144) = 12.

Tell me which quantities are labeled (or upload the picture) and I’ll compute the height.