Pyramids and Cones — Hard Practice Quiz

A Geometry cheat sheet for Pyramids and Cones — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. If a cone's height is doubled and its radius is halved, how does its volume change?

   • It remains the same.
   • It is doubled.
   • It is halved.
   • It is quadrupled.

   Answer: It is halved.

2. Which of the following equations correctly relates the height $h$, volume $V$, and slant height $l$ of a cone?

   • $h^3 - l^2 h + \frac{3V}{\pi} = 0$
   • $h^3 + l^2 h - \frac{3V}{\pi} = 0$
   • $h^2 - l^2 h + \frac{3V}{\pi} = 0$
   • $h^3 - l h^2 + \frac{3V}{\pi} = 0$

   Answer: $h^3 - l^2 h + \frac{3V}{\pi} = 0$

3. A sector of a circle with radius $R$ is rolled to form a cone. If the cone's height is $h$, what is its volume?

   • $V = \frac{1}{3}\pi (R^2 - h^2) h$
   • $V = \frac{1}{3}\pi R^2 h$
   • $V = \frac{1}{3}\pi (R^2 + h^2) h$
   • $V = \frac{1}{3}\pi (R - h)^2 h$

   Answer: $V = \frac{1}{3}\pi (R^2 - h^2) h$

4. Two cones have the same volume. If the radius of the first cone is twice the radius of the second cone, what is the ratio of the height of the first cone to the height of the second cone?

   • $\frac{1}{2}$
   • $\frac{1}{4}$
   • $2$
   • $4$

   Answer: $\frac{1}{4}$

5. A cone has a volume of $V$ and a height of $h$. What is its slant height $l$?

   • $l = \sqrt{\frac{3V}{\pi h} + h^2}$
   • $l = \sqrt{\frac{V}{\pi h} + h^2}$
   • $l = \sqrt{\frac{3V}{\pi h} - h^2}$
   • $l = \frac{3V}{\pi h} + h$

   Answer: $l = \sqrt{\frac{3V}{\pi h} + h^2}$

6. If all linear dimensions (radius and height) of a cone are scaled by a factor of $k$, how do its volume and surface area change?

   • Volume scales by $k^2$, Surface Area scales by $k^3$.
   • Volume scales by $k^3$, Surface Area scales by $k^2$.
   • Volume scales by $k$, Surface Area scales by $k^2$.
   • Volume scales by $k^3$, Surface Area scales by $k$.

   Answer: Volume scales by $k^3$, Surface Area scales by $k^2$.

7. Given the surface area $SA$ and slant height $l$ of a cone, which expression correctly represents its radius $r$?

   • $r = \frac{-\pi l + \sqrt{\pi^2 l^2 + 4\pi SA}}{2\pi}$
   • $r = \frac{\pi l + \sqrt{\pi^2 l^2 + 4\pi SA}}{2\pi}$
   • $r = \frac{-\pi l - \sqrt{\pi^2 l^2 + 4\pi SA}}{2\pi}$
   • $r = \frac{-\pi l + \sqrt{\pi^2 l^2 - 4\pi SA}}{2\pi}$

   Answer: $r = \frac{-\pi l + \sqrt{\pi^2 l^2 + 4\pi SA}}{2\pi}$

8. A pyramid has a square base with side length $s$ and a slant height $L$ for its triangular faces. Express its volume in terms of $s$ and $L$.

   • $V = \frac{1}{3} s^2 \sqrt{L^2 - \frac{s^2}{4}}$
   • $V = \frac{1}{3} s^2 \sqrt{L^2 + \frac{s^2}{4}}$
   • $V = \frac{1}{3} s^2 \sqrt{L^2 - s^2}$
   • $V = \frac{1}{3} s^2 L$

   Answer: $V = \frac{1}{3} s^2 \sqrt{L^2 - \frac{s^2}{4}}$

9. A cone has a surface area $SA$. If its height is equal to its radius ($h=r$), express its volume in terms of $SA$ and $\pi$.

   • $V = \frac{1}{3}\pi \left( \frac{SA}{\pi (1 + \sqrt{2})} \right)^{3/2}$
   • $V = \frac{1}{3}\pi \left( \frac{SA}{\pi (1 + \sqrt{2})} \right)^{1/2}$
   • $V = \frac{1}{3}\pi \left( \frac{SA}{\pi (1 + \sqrt{2})} \right)$
   • $V = \frac{SA}{3(1 + \sqrt{2})}$

   Answer: $V = \frac{1}{3}\pi \left( \frac{SA}{\pi (1 + \sqrt{2})} \right)^{3/2}$

10. A cone has a slant height $l$ that is twice its height $h$. What is the ratio of its volume to its surface area?

    • $\frac{h}{3 + 2\sqrt{3}}$
    • $\frac{h}{1 + \sqrt{3}}$
    • $\frac{h}{3 + \sqrt{3}}$
    • $\frac{h}{2 + 3\sqrt{3}}$

    Answer: $\frac{h}{3 + 2\sqrt{3}}$