Radians and Degrees
The one fact everything comes from
$180^\circ = \pi$ radians. To change degrees to radians, multiply by $\dfrac{\pi}{180}$. To change radians to degrees, multiply by $\dfrac{180}{\pi}$.
Radians matter most for arcs. A central angle of $\theta$ radians in a circle of radius $r$ cuts off an arc of length $r\theta$, and that formula only works when $\theta$ is in radians.
Radian questions on the SAT ask you to convert between degrees and radians, to find arc lengths and sector areas, to find an angle from an arc, and to read points on the unit circle at angles like $\dfrac{2\pi}{3}$.
What to do with each type
The SAT reference sheet reminds you that a full circle is $360^\circ$ or $2\pi$ radians. These are the angles that come up again and again. Knowing them saves a conversion every time.
- $30^\circ$. $\dfrac{\pi}{6}$
$135^\circ$ $\dfrac{3\pi}{4}$ - $45^\circ$. $\dfrac{\pi}{4}$
$150^\circ$ $\dfrac{5\pi}{6}$ - $60^\circ$. $\dfrac{\pi}{3}$
$180^\circ$ $\pi$ - $90^\circ$. $\dfrac{\pi}{2}$
$270^\circ$ $\dfrac{3\pi}{2}$ - $120^\circ$. $\dfrac{2\pi}{3}$
$360^\circ$ $2\pi$ A central angle of $\theta$ radians cuts off an arc of length $s = r\theta$. One radian is the angle whose arc is exactly one radius long, about $57.3^\circ$.
- Arc length. $s = r\theta$
$s = \dfrac{n}{360} \cdot 2\pi r$ - Sector area. $A = \dfrac{1}{2}r^2\theta$
$A = \dfrac{n}{360} \cdot \pi r^2$ Examples, step by step
Example 1: radians to degrees
An angle measures $\dfrac{5\pi}{9}$ radians. What is its measure in degrees?
- Multiply by $\dfrac{180}{\pi}$: $\dfrac{5\pi}{9} \cdot \dfrac{180}{\pi}$.
- The $\pi$ cancels: $\dfrac{5 \cdot 180}{9} = 5 \cdot 20 = 100$.
- The angle is $100^\circ$. A quick check: $\dfrac{5}{9}$ of $\pi$ is a little more than half of $180^\circ$.
Example 2: degrees to radians
Convert $225^\circ$ to radians.
- Multiply by $\dfrac{\pi}{180}$: $225 \cdot \dfrac{\pi}{180} = \dfrac{225\pi}{180}$.
- Both numbers divide by $45$: $\dfrac{5\pi}{4}$.
- Leave $\pi$ in the answer unless the question asks for a decimal.
Example 3: arc length
A circle has radius $6$. A central angle of $\dfrac{2\pi}{3}$ radians cuts off an arc. How long is the arc?
- The angle is already in radians, so use $s = r\theta$.
- $s = 6 \cdot \dfrac{2\pi}{3} = 4\pi$.
- Check with degrees: $\dfrac{2\pi}{3} = 120^\circ$, a third of the circle, and a third of the circumference $12\pi$ is $4\pi$.
Example 4: the angle from the arc
An arc of length $10$ is on a circle of radius $4$. What is the central angle in radians?
- Rearrange $s = r\theta$ to get $\theta = \dfrac{s}{r}$.
- $\theta = \dfrac{10}{4} = 2.5$ radians.
- That is about $143^\circ$. Notice there is no $\pi$ here. A radian measure does not have to include $\pi$.
Common mistakes
- Using the wrong fraction. Degrees to radians uses $\dfrac{\pi}{180}$. Radians to degrees uses $\dfrac{180}{\pi}$. If $\pi$ is not cancelling when you go to degrees, you used the wrong one.
- Putting degrees into $s = r\theta$. This formula only works with radians. With $120^\circ$, convert first, or use $\dfrac{n}{360} \cdot 2\pi r$.
- Thinking a full circle is $\pi$ radians. A full circle is $2\pi$ radians. $\pi$ radians is a half circle.
- Treating $\theta = 2.5$ as $2.5\pi$. A radian measure can be a plain number. $2.5$ radians is about $143^\circ$, while $2.5\pi$ radians is $450^\circ$.
- Forgetting the $\dfrac{1}{2}$ in sector area. With radians, the sector area is $\dfrac{1}{2}r^2\theta$, not $r^2\theta$.
Practice
Ten questions, from easy to hard. Keep $180^\circ = \pi$ in mind for every one of them.
EasyAn angle measures $\dfrac{5\pi}{9}$ radians. What is the measure of the angle in degrees?- A$50^\circ$
- B$100^\circ$
- C$162^\circ$
- D$900^\circ$
Show solution
Answer: B. $\dfrac{5\pi}{9} \cdot \dfrac{180}{\pi} = \dfrac{900}{9} = 100$. The angle is $100^\circ$. Choice D forgets to divide by $9$.EasyWhich of the following is equal to $135^\circ$?- A$\dfrac{2\pi}{3}$ radians
- B$\dfrac{4\pi}{3}$ radians
- C$\dfrac{3\pi}{4}$ radians
- D$\dfrac{3\pi}{2}$ radians
Show solution
Answer: C. $135 \cdot \dfrac{\pi}{180} = \dfrac{135\pi}{180} = \dfrac{3\pi}{4}$, after dividing both numbers by $45$.EasyStudent-produced responseAn angle measures $\dfrac{\pi}{12}$ radians. What is the measure of the angle in degrees?Enter your answer.Show solution
Answer: $15$. $\dfrac{\pi}{12} \cdot \dfrac{180}{\pi} = \dfrac{180}{12} = 15$.MediumA circle has a radius of $6$. A central angle of $\dfrac{2\pi}{3}$ radians cuts off an arc of the circle. What is the length of the arc?- A$2\pi$
- B$4$
- C$12\pi$
- D$4\pi$
Show solution
Answer: D. $s = r\theta = 6 \cdot \dfrac{2\pi}{3} = 4\pi$. Choice C is the whole circumference.MediumStudent-produced responseAn arc of length $10$ lies on a circle of radius $4$. What is the measure, in radians, of the central angle that cuts off the arc?Enter your answer.Show solution
Answer: $2.5$ (or $\dfrac{5}{2}$). $\theta = \dfrac{s}{r} = \dfrac{10}{4} = 2.5$ radians.MediumA sector of a circle with radius $10$ has a central angle of $\dfrac{\pi}{5}$ radians. What is the area of the sector?- A$10\pi$
- B$2\pi$
- C$20\pi$
- D$100\pi$
Show solution
Answer: A. $A = \dfrac{1}{2}r^2\theta = \dfrac{1}{2}(100)\left(\dfrac{\pi}{5}\right) = 10\pi$. Choice C forgets the $\dfrac{1}{2}$, and choice D is the whole circle.MediumThe minute hand of a clock turns for $20$ minutes. Through what angle, in radians, does it turn?- A$\dfrac{\pi}{3}$
- B$\dfrac{2\pi}{3}$
- C$\dfrac{\pi}{20}$
- D$20\pi$
Show solution
Answer: B. $20$ minutes is $\dfrac{20}{60} = \dfrac{1}{3}$ of a full turn. A full turn is $2\pi$ radians, so the angle is $\dfrac{1}{3} \cdot 2\pi = \dfrac{2\pi}{3}$, which is $120^\circ$. Choice A uses $\pi$ for a full turn.HardA wheel with a radius of $15$ inches rotates through an angle of $240^\circ$. How far does a point on the edge of the wheel travel, in inches?- A$10\pi$
- B$20\pi$
- C$40\pi$
- D$3{,}600\pi$
Show solution
Answer: B. Convert first: $240 \cdot \dfrac{\pi}{180} = \dfrac{4\pi}{3}$. Then $s = r\theta = 15 \cdot \dfrac{4\pi}{3} = 20\pi$, about $62.8$ inches. Choice D comes from putting $240$ straight into $s = r\theta$.HardStudent-produced responseAn angle has a measure of $\dfrac{3\pi}{k}$ radians, where $k$ is a positive constant. The same angle measures $54^\circ$. What is the value of $k$?Enter your answer.Show solution
Answer: $10$. Convert the radian measure to degrees: $\dfrac{3\pi}{k} \cdot \dfrac{180}{\pi} = \dfrac{540}{k}$. Set $\dfrac{540}{k} = 54$, so $k = 10$.HardIn the $xy$-plane, point $P$ lies on the circle $x^2 + y^2 = 1$. The ray from the origin through $P$ makes an angle of $\dfrac{2\pi}{3}$ radians, measured counterclockwise from the positive $x$-axis. What are the coordinates of $P$?- A$\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$
- B$\left(-\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$
- C$\left(-\dfrac{1}{2}, -\dfrac{\sqrt{3}}{2}\right)$
- D$\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$
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Answer: D. $\dfrac{2\pi}{3}$ radians is $120^\circ$, which is in the second quadrant, so $x$ is negative and $y$ is positive. The reference angle is $60^\circ$, and the coordinates are $(\cos 120^\circ, \sin 120^\circ) = \left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$. Choice A is the point at $60^\circ$.Is this a topic you need?
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