Spheres — Hard Practice Quiz

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

Practice quiz

1. If the radius of a sphere is doubled, how does the ratio of its volume to its surface area change?

   • It becomes $2$ times larger.
   • It becomes $4$ times larger.
   • It becomes $8$ times larger.
   • It remains unchanged.

   Answer: It becomes $8$ times larger.

2. A spherical cap is removed from a sphere. If the cap's height $h$ is exactly half the sphere's radius $r$, what fraction of the sphere's total volume is the cap's volume?

   • $\frac{1}{8}$
   • $\frac{1}{4}$
   • $\frac{5}{32}$
   • $\frac{3}{16}$

   Answer: $\frac{5}{32}$

3. A spherical cap is removed from a sphere. If the cap's height $h$ is exactly one-fourth of the sphere's radius $r$, what fraction of the sphere's total surface area is the cap's surface area?

   • $\frac{1}{2}$
   • $\frac{1}{4}$
   • $\frac{1}{8}$
   • $\frac{1}{16}$

   Answer: $\frac{1}{8}$

4. Given the volume of a spherical cap $V_{cap}$ and its height $h$, derive an expression for the radius $r$ of the original sphere in terms of $V_{cap}$, $h$, and $\pi$.

   • $r = \frac{V_{cap}}{\pi h^2} + \frac{h}{3}$
   • $r = \frac{3V_{cap}}{\pi h^2} - \frac{h}{3}$
   • $r = \frac{V_{cap}}{3\pi h^2} + h$
   • $r = \frac{3V_{cap}}{h^2} + \frac{h}{\pi}$

   Answer: $r = \frac{V_{cap}}{\pi h^2} + \frac{h}{3}$

5. A sphere of radius $r$ is cut by two parallel planes, creating two spherical caps with heights $h_1$ and $h_2$. What is the total volume of these two spherical caps?

   • $\frac{1}{3}\pi (h_1^2(3r - h_1) + h_2^2(3r - h_2))$
   • $\frac{1}{3}\pi (h_1 + h_2)^2(3r - (h_1 + h_2))$
   • $\frac{4}{3}\pi r^3 - \frac{1}{3}\pi (h_1 + h_2)^2(3r - (h_1 + h_2))$
   • $\frac{1}{3}\pi (h_1^2(3r - h_1) - h_2^2(3r - h_2))$

   Answer: $\frac{1}{3}\pi (h_1^2(3r - h_1) + h_2^2(3r - h_2))$

6. A sphere of radius $r$ is cut by two parallel planes, creating two spherical caps with heights $h_1$ and $h_2$. What is the curved surface area of the remaining central frustum?

   • $2\pi r (h_1 + h_2)$
   • $4\pi r^2 - 2\pi r (h_1 + h_2)$
   • $2\pi r (2r - h_1 - h_2)$
   • $4\pi r^2 - 2\pi r h_1 h_2$

   Answer: $2\pi r (2r - h_1 - h_2)$

7. A spherical cap has a surface area $SA_{cap}$ and a height $h$. What is the volume of the original sphere from which this cap was cut, in terms of $SA_{cap}$, $h$, and $\pi$?

   • $\frac{SA_{cap}^3}{6\pi^2 h^3}$
   • $\frac{SA_{cap}^3}{24\pi^2 h^3}$
   • $\frac{4SA_{cap}^3}{3\pi h^3}$
   • $\frac{SA_{cap}^3}{3\pi h^3}$

   Answer: $\frac{SA_{cap}^3}{6\pi^2 h^3}$

8. For a spherical cap with height $h$ and radius $r$ of the original sphere, what is the ratio of its volume to its surface area?

   • $\frac{h(3r - h)}{6r}$
   • $\frac{h(3r - h)}{2r}$
   • $\frac{h(3r - h)}{3r}$
   • $\frac{h^2(3r - h)}{6r}$

   Answer: $\frac{h(3r - h)}{6r}$

9. If the radius $r$ of a sphere is doubled, but the height $h$ of a spherical cap cut from it remains constant, what is the new volume of the spherical cap?

   • $\frac{1}{3}\pi h^2(6r - h)$
   • $\frac{1}{3}\pi h^2(3r - h)$
   • $\frac{1}{3}\pi h^2(3r - 2h)$
   • $\frac{1}{3}\pi (2h)^2(6r - 2h)$

   Answer: $\frac{1}{3}\pi h^2(6r - h)$

10. A sphere has a total surface area $SA$. If a spherical cap is removed such that its surface area $SA_{cap}$ is exactly one-quarter of the sphere's total surface area, what is the height $h$ of the cap in terms of the sphere's radius $r$?

    • $h = r$
    • $h = \frac{r}{2}$
    • $h = \frac{r}{4}$
    • $h = 2r$

    Answer: $h = \frac{r}{2}$