Circles — Hard Practice Quiz

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

Practice quiz

1. A circular sector has an arc length $L_{arc}$ and a radius $r$. Which of the following expressions correctly represents the area of the circular segment formed by this sector?

   • $\frac{1}{2}r L_{arc} - \frac{1}{2}r^2 \sin(\frac{L_{arc}}{r})$
   • $\frac{1}{2}r^2(\frac{L_{arc}}{r} - \cos(\frac{L_{arc}}{r}))$
   • $\frac{1}{2}r L_{arc} - r \sin(\frac{L_{arc}}{r})$
   • $\frac{1}{2}r^2(\frac{L_{arc}}{r} - \tan(\frac{L_{arc}}{r}))$

   Answer: $\frac{1}{2}r L_{arc} - \frac{1}{2}r^2 \sin(\frac{L_{arc}}{r})$

2. If the arc length of a circular sector is numerically equal to its radius $r$, what is the ratio of the sector's area to the total area of the circle?

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

   Answer: $\frac{1}{2\pi}$

3. A circle has a circumference of $10\pi \text{ cm}$. A sector of this circle has an area of $15\pi \text{ cm}^2$. What is the arc length of this sector?

   • $6\pi \text{ cm}$
   • $5\pi \text{ cm}$
   • $10\pi \text{ cm}$
   • $3\pi \text{ cm}$

   Answer: $6\pi \text{ cm}$

4. If the radius of a circle is increased by $20\%$ and the central angle of a sector within it is decreased by $10\%$, what is the percentage change in the sector's area?

   • $29.6\%$ increase
   • $20\%$ increase
   • $10\%$ increase
   • $15.2\%$ increase

   Answer: $29.6\%$ increase

5. A circular segment is defined by an arc whose length is equal to the radius $r$ of the circle. What is the area of this segment?

   • $\frac{1}{2}r^2(1 - \sin(1))$
   • $\frac{1}{2}r^2(1 - \cos(1))$
   • $\frac{1}{2}r^2(\pi - \sin(\pi))$
   • $\frac{1}{2}r^2(1 - \frac{\pi}{2})$

   Answer: $\frac{1}{2}r^2(1 - \sin(1))$

6. A circular sector has an area $A_{sector}$ and an arc length $L_{arc}$. Which expression correctly gives the radius $r$ of the circle?

   • $\frac{2A_{sector}}{L_{arc}}$
   • $\frac{A_{sector}}{2L_{arc}}$
   • $\frac{L_{arc}}{2A_{sector}}$
   • $\frac{A_{sector}L_{arc}}{2}$

   Answer: $\frac{2A_{sector}}{L_{arc}}$

7. For what central angle $\theta$ (in radians, $0 < \theta \le 2\pi$) is the area of a circular sector equal to the area of the circular segment it defines?

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

   Answer: $\pi$

8. A circle has a circumference of $12\pi \text{ cm}$. A segment of this circle has an arc length of $4\pi \text{ cm}$. What is the area of this segment?

   • $(12\pi - 9\sqrt{3}) \text{ cm}^2$
   • $(12\pi - 18\sqrt{3}) \text{ cm}^2$
   • $(6\pi - 9\sqrt{3}) \text{ cm}^2$
   • $(18\pi - 9\sqrt{3}) \text{ cm}^2$

   Answer: $(12\pi - 9\sqrt{3}) \text{ cm}^2$

9. Two concentric circles have arc lengths $L_1 = 3 \text{ cm}$ and $L_2 = 6 \text{ cm}$ for the same central angle. If the area of the larger sector is $18 \text{ cm}^2$, what is the area of the region between the two arcs?

   • $13.5 \text{ cm}^2$
   • $9 \text{ cm}^2$
   • $12 \text{ cm}^2$
   • $15 \text{ cm}^2$

   Answer: $13.5 \text{ cm}^2$

10. For what central angle $\theta$ (in radians, $0 < \theta < 2\pi$) is the area of a circular segment exactly half the area of its corresponding sector?

    • The solution to $\frac{1}{2}\theta = \sin\theta$
    • $\frac{\pi}{3}$
    • $\frac{\pi}{2}$
    • The solution to $\theta = 2\sin\theta$

    Answer: The solution to $\frac{1}{2}\theta = \sin\theta$