Radian and Degree Measures — Hard Practice Quiz

A Trigonometry cheat sheet for Radian and Degree Measures — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Radians to Degrees

$$1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$$

Degrees to Radians

$$1^\circ = \frac{\pi}{180} \text{ rad} \approx 0.0175 \text{ rad}$$

Practice quiz

  1. A circle has a radius of $15 \text{ cm}$. An arc subtends a central angle of $108^\circ$. What is the length of the arc?

    • $9\pi \text{ cm}$
    • $18\pi \text{ cm}$
    • $108\pi \text{ cm}$
    • $27\pi \text{ cm}$

    Answer: $9\pi \text{ cm}$

  2. A sector of a circle has a radius of $8 \text{ m}$ and an area of $16\pi \text{ m}^2$. What is the central angle of the sector in degrees?

    • $45^\circ$
    • $90^\circ$
    • $180^\circ$
    • $360^\circ$

    Answer: $90^\circ$

  3. A wheel rotates at a constant angular speed. In $10 \text{ seconds}$, it completes $5$ full revolutions. What is its angular speed in degrees per second?

    • $36^\circ/\text{s}$
    • $180^\circ/\text{s}$
    • $360^\circ/\text{s}$
    • $540^\circ/\text{s}$

    Answer: $180^\circ/\text{s}$

  4. Angle $X$ is $2.5 \text{ radians}$. Angle $Y$ is $135^\circ$. What is the ratio of angle $X$ to angle $Y$ when both are expressed in the same unit?

    • $\frac{10}{3\pi}$
    • $\frac{3\pi}{10}$
    • $\frac{5}{3\pi}$
    • $\frac{10}{\pi}$

    Answer: $\frac{10}{3\pi}$

  5. If an angle $\theta$ satisfies $\cos(\theta) = \frac{\sqrt{3}}{2}$ and $\theta$ is in the first quadrant, what is the value of $\theta$ in radians?

    • $\frac{\pi}{3}$
    • $\frac{\pi}{4}$
    • $\frac{\pi}{6}$
    • $\frac{\pi}{2}$

    Answer: $\frac{\pi}{6}$

  6. A pendulum swings through an angle of $A \text{ degrees}$. If the length of the arc traced by the pendulum bob is $L$, and the length of the pendulum is $R$, which of the following expressions correctly relates $L$, $R$, and $A$?

    • $L = R A$
    • $L = R A \frac{\pi}{180}$
    • $L = R A \frac{180}{\pi}$
    • $L = \frac{1}{2} R^2 A$

    Answer: $L = R A \frac{\pi}{180}$

  7. A circular track has a radius of $200 \text{ m}$. A runner completes a segment of the track that subtends a central angle of $144^\circ$. If the runner maintains a constant speed of $4 \text{ m/s}$, how long does it take to complete this segment?

    • $20\pi \text{ s}$
    • $40\pi \text{ s}$
    • $80\pi \text{ s}$
    • $160\pi \text{ s}$

    Answer: $40\pi \text{ s}$

  8. Object A rotates at $60 \text{ revolutions per minute}$. Object B rotates at $3\pi \text{ rad/s}$. Which object has a greater angular speed, and by how much (in degrees per second)?

    • Object A by $180^\circ/\text{s}$
    • Object B by $180^\circ/\text{s}$
    • Object A by $270^\circ/\text{s}$
    • Object B by $270^\circ/\text{s}$

    Answer: Object B by $180^\circ/\text{s}$

  9. An angle $\alpha$ is given in radians, and another angle $\beta$ is given in degrees. If $\alpha = k \beta$, where $k$ is a constant, what is the value of $k$ such that $\alpha$ and $\beta$ represent the same physical angle?

    • $\frac{180}{\pi}$
    • $\frac{\pi}{180}$
    • $1$
    • $\pi$

    Answer: $\frac{\pi}{180}$

  10. A regular hexagon is inscribed in a circle. What is the central angle subtended by one side of the hexagon, expressed in radians?

    • $\frac{\pi}{6}$
    • $\frac{\pi}{4}$
    • $\frac{\pi}{3}$
    • $\frac{2\pi}{3}$

    Answer: $\frac{\pi}{3}$

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