Radian Measure and Arc Length - Precalculus

1. What is the radian-to-degree conversion formula for an angle measuring $x$ radians?

   $x\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ since $\pi$ radians $= 180°$.

2. Find the radius $r$ when arc length $s=10\pi$ and $\theta=\frac{5\pi}{2}$ (radians).

   $4$. Use $r=\frac{s}{\theta}$: $r=\frac{10\pi}{\frac{5\pi}{2}}=4$.

3. What is the area of a full circle written to match the sector area formula $A=\frac{1}{2}r^2\theta$?

   $A=\pi r^2$. Full circle has $\theta = 2\pi$, so $A = \frac{1}{2}r^2(2\pi) = \pi r^2$.

4. Find the central angle $\theta$ in radians when arc length $s=12$ and radius $r=3$.

   $4$. Use $\theta=\frac{s}{r}$: $\theta=\frac{12}{3}=4$.

5. Find the arc length $s$ when $r=5$ and $\theta=\frac{3\pi}{2}$ (radians).

   $\frac{15\pi}{2}$. Use $s=r\theta$: $s=5\cdot\frac{3\pi}{2}=\frac{15\pi}{2}$.

6. Convert $150^\circ$ to radians.

   $\frac{5\pi}{6}$. $150 \cdot \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}$.

7. What is the circumference formula $C$ of a circle in terms of radius $r$?

   $C=2\pi r$. Circumference is $2\pi$ times the radius.

8. What is the circumference of a circle written using the arc length formula $s=r\theta$?

   $C=2\pi r$. Full circle has angle $2\pi$, so $s = r(2\pi) = 2\pi r$.

9. Find the sector area $A$ when $r=6$ and $\theta=\frac{\pi}{3}$ (radians).

   $6\pi$. Use $A=\frac{1}{2}r^2\theta$: $A=\frac{1}{2}(36)(\frac{\pi}{3})=6\pi$.

10. What is the area formula $A$ of a circle in terms of radius $r$?

    $A=\pi r^2$. Circle area is $\pi$ times radius squared.

11. What is the degree-to-radian conversion formula for an angle measuring $x^\circ$?

    $x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ since $180° = \pi$ radians.

12. What is the radian measure of a full rotation (one complete circle)?

    $2\pi$. Full circle's arc length equals circumference: $\frac{2\pi r}{r} = 2\pi$.

13. What is the arc length formula for a circle of radius $r$ with central angle $\theta$ measured in radians?

    $s=r\theta$. Multiply radius by angle in radians to get arc length.

14. Convert $\frac{7\pi}{4}$ radians to degrees.

    $315^\circ$. $\frac{7\pi}{4} \cdot \frac{180}{\pi} = \frac{7 \cdot 180}{4} = 315$.

15. What is the radian-to-degree conversion formula for an angle of $x$ radians?

    $x\cdot\frac{180}{\pi}$. Multiply radians by $\frac{180}{\pi}$ to get degrees.

16. Find the arc length $s$ if $r=5$ and $\theta=\frac{3\pi}{2}$ radians.

    $s=\frac{15\pi}{2}$. Using $s = r\theta$: $s = 5 \cdot \frac{3\pi}{2} = \frac{15\pi}{2}$.

17. Identify the central angle $\theta$ in radians if the arc length equals the radius, $s=r$.

    $1$. When $s=r$, then $\theta=\frac{s}{r}=1$ radian.

18. Identify the correct arc length if $\theta$ is in degrees: which formula is correct for $s$?

    $s=\frac{\theta}{360}\cdot 2\pi r$. Fraction $\frac{\theta}{360}$ of circumference $2\pi r$ gives arc length.

19. What is the definition of radian measure $\theta$ using arc length $s$ and radius $r$?

    $\theta=\frac{s}{r}$. Radian measure is the ratio of arc length to radius.

20. What is the radian measure of $180^\circ$?

    $\pi$. 180° equals $\pi$ radians (half circle).

21. What does it mean to say arc length is proportional to radius for a fixed central angle $\theta$?

    $\frac{s}{r}$ is constant for that $\theta$. The ratio of arc length to radius remains the same for any given angle.

22. What is the sector area formula written using arc length $s$ and radius $r$?

    $A=\frac{1}{2}rs$. Substitute $s = r\theta$ into $A = \frac{1}{2}r^2\theta$.

23. Find the central angle $\theta$ in radians if arc length $s=12$ and radius $r=3$.

    $\theta=4$. Using $\theta = \frac{s}{r}$: $\theta = \frac{12}{3} = 4$.

24. What is the area of a sector with radius $r$ and central angle $\theta$ measured in radians?

    $A=\frac{1}{2}r^2\theta$. Sector is fraction $\frac{\theta}{2\pi}$ of circle area $\pi r^2$.

25. Find the radius $r$ if arc length $s=10\pi$ and central angle $\theta=5$ radians.

    $r=2\pi$. Using $r = \frac{s}{\theta}$: $r = \frac{10\pi}{5} = 2\pi$.

26. Find the sector area $A$ using $A=\frac{1}{2}rs$ if $r=8$ and $s=6\pi$.

    $A=24\pi$. Direct substitution: $A = \frac{1}{2}(8)(6\pi) = 24\pi$.

27. What is the degree-to-radian conversion formula for an angle of $x^\circ$?

    $x^\circ\cdot\frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to get radians.

28. What is the definition of radian measure of a central angle $\theta$ in terms of arc length $s$ and radius $r$?

    $\theta=\frac{s}{r}$. Radian measure equals arc length divided by radius.

29. What is the arc length formula $s$ for a central angle $\theta$ measured in radians in a circle of radius $r$?

    $s=r\theta$. Arc length equals radius times angle in radians.

30. Find the sector area $A$ if $r=6$ and $\theta=\frac{\pi}{3}$ radians.

    $A=6\pi$. Using $A = \frac{1}{2}r^2\theta$: $A = \frac{1}{2}(36)\frac{\pi}{3} = 6\pi$.

31. What is the sector area formula $A$ for a central angle $\theta$ in radians and radius $r$?

    $A=\frac{1}{2}r^2\theta$. Sector area is half the product of radius squared and angle.

32. What is the radian measure of a straight angle (a semicircle)?

    $\pi$. Half circle is half of $2\pi$ radians.

33. What is the radian measure of a full revolution (one complete circle)?

    $2\pi$. A full circle spans $2\pi$ radians (360°).

34. Find the sector area $A$ when $r=4$ and arc length $s=6$ (use $A=\frac{1}{2}rs$).

    $12$. Use $A=\frac{1}{2}rs$: $A=\frac{1}{2}(4)(6)=12$.

35. What is the radian measure of a right angle?

    $\frac{\pi}{2}$. Quarter circle is one-fourth of $2\pi$ radians.

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