3D Shapes - Advanced — Hard Practice Quiz

A Geometry cheat sheet for 3D Shapes - Advanced — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. A cube has a space diagonal equal to the edge length of a regular tetrahedron. If the cube's volume is $V_C$, what is the tetrahedron's volume $V_T$ in terms of $V_C$?

   • $ \frac{\sqrt{6}}{4}V_C $
   • $ \frac{\sqrt{3}}{2}V_C $
   • $ \frac{\sqrt{2}}{3}V_C $
   • $ \frac{1}{2}V_C $

   Answer: $ \frac{\sqrt{6}}{4}V_C $

2. A torus has a major radius $R$ and minor radius $r$. If its volume is $V$ and surface area is $SA$, express $R$ in terms of $V$, $SA$, and $\pi$.

   • $ R = \frac{SA^2}{8\pi^2 V} $
   • $ R = \frac{V SA}{2\pi^2} $
   • $ R = \frac{SA}{2\pi r} $
   • $ R = \frac{V}{2\pi^2 r^2} $

   Answer: $ R = \frac{SA^2}{8\pi^2 V} $

3. An ellipsoid has semi-axes $a=2x$, $b=3x$, $c=x$. If its volume is equal to the volume of a cube with edge length $y$, what is $x$ in terms of $y$ and $\pi$?

   • $ x = \frac{y}{2\pi^{1/3}} $
   • $ x = \frac{y}{(6\pi)^{1/3}} $
   • $ x = \frac{y}{2\sqrt[3]{\pi}} $
   • $ x = \frac{y}{4\pi^{1/3}} $

   Answer: $ x = \frac{y}{2\pi^{1/3}} $

4. A frustum of a pyramid has base areas $B_1$ and $B_2$. If $B_1$ is the surface area of a cube with edge length $L$, and $B_2$ is the surface area of a cube with edge length $2L$, and the frustum's height is $3L$, what is the frustum's volume in terms of $L$?

   • $ 42L^3 $
   • $ 36L^3 $
   • $ 54L^3 $
   • $ 28L^3 $

   Answer: $ 42L^3 $

5. A torus has a major radius $R=2a$ and minor radius $r=a$. An ellipsoid has semi-axes $a, b=a, c=a$. What is the ratio of the torus volume to the ellipsoid volume?

   • $ 3\pi $
   • $ 2\pi $
   • $ 4\pi $
   • $ \frac{3}{2}\pi $

   Answer: $ 3\pi $

6. A cube has a surface area $SA_C$. A torus has a major radius $R = \frac{SA_C}{24\pi}$ and minor radius $r = \frac{SA_C}{12\pi}$. What is the surface area of the torus $SA_T$ in terms of $SA_C$?

   • $ \frac{SA_C^2}{72} $
   • $ \frac{SA_C^2}{144} $
   • $ \frac{SA_C^2}{36} $
   • $ \frac{SA_C^2}{24\pi} $

   Answer: $ \frac{SA_C^2}{72} $

7. A regular tetrahedron has edge length $a$. A frustum of a pyramid has base areas $B_1 = a^2$ and $B_2 = 4a^2$. If the frustum's height is such that its volume is equal to the tetrahedron's volume, what is the height $h$ of the frustum in terms of $a$?

   • $ \frac{a\sqrt{2}}{28} $
   • $ \frac{a\sqrt{2}}{14} $
   • $ \frac{a\sqrt{2}}{7} $
   • $ \frac{a\sqrt{2}}{42} $

   Answer: $ \frac{a\sqrt{2}}{28} $

8. If the major radius $R$ of a torus is doubled and the minor radius $r$ is halved, how does its volume $V$ and surface area $SA$ change?

   • Volume is halved, Surface Area remains the same.
   • Volume remains the same, Surface Area is halved.
   • Both Volume and Surface Area are halved.
   • Volume is doubled, Surface Area remains the same.

   Answer: Volume is halved, Surface Area remains the same.

9. An ellipsoid has semi-axes $a=2$, $b=3$, $c=4$. A frustum has base areas $B_1 = 4\pi$ and $B_2 = 9\pi$ and height $h=6$. What is the ratio of the ellipsoid's volume to the frustum's volume?

   • $ \frac{16}{19} $
   • $ \frac{32}{38} $
   • $ \frac{1}{2} $
   • $ \frac{2}{3} $

   Answer: $ \frac{16}{19} $

10. A cube has a space diagonal $d$. A torus has a major radius $R = d$ and minor radius $r = \frac{d}{2\pi}$. What is the ratio of the cube's surface area to the torus's surface area?

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

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