Random point in the lower half of a cone

Question

A cone has a circular base of radius $1$ and height $2$ . A point is chosen uniformly from the solid cone. What is the probability that its distance above the base is no more than $1$ ?

Short answer

$\tfrac78$ . The part of the cone above height 1 is a similar cone scaled by $\tfrac12$ , so it has $\left(\tfrac12\right)^3=\tfrac18$ of the volume. The remaining $\tfrac78$ lies within height 1 of the base.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Continuous Random Variables and Probability Density Functions
Cumulative Distribution Functions

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