Radians and Degrees
Angles can be measured in degrees or radians. The anchor: a straight angle is $180° = \pi$ radians.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
Degrees to radians: multiply by $\frac{\pi}{180}$. Radians to degrees: multiply by $\frac{180}{\pi}$. Pick the direction that cancels the unit you're starting with.
How to solve one, step by step
Example: convert $60°$ to radians.
- Multiply by $\frac{\pi}{180}$: $60 \cdot \frac{\pi}{180}$.
- Simplify: $\frac{\pi}{3}$.
The mistakes that cost points
- Converting in the wrong direction. Use $\frac{\pi}{180}$ for degrees-to-radians and $\frac{180}{\pi}$ the other way.
- Confusing radians with arc length. A radian is an angle measure, not a length along the circle.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A given angle measures $\frac{5\pi}{9}\text{ radians}$. What is the equivalent measure in degrees?
- A$\frac{\pi^2}{324}°$
- B$20°$
- C$100°$
- D$\frac{5}{9}°$
Show solution
Answer: C, $100°$. To convert radians to degrees, multiply by $\frac{180}{\pi}$. $\frac{5\pi}{9} \cdot \frac{180}{\pi} = 100°$. Common error: multiplying by $\frac{\pi}{180}$ (the wrong direction).
Medium
The figure shows angle $A$ (an obtuse angle) and angle $B$ (a negative acute angle), formed by the same straight line and the positive x-axis. Suppose angle $B$ measures $-\frac{3\pi}{10}$ radians. What is the measure of angle $A$ in degrees? Note: figure not drawn to scale.
- A$-54°$
- B$-126°$
- C$126°$
- D$54°$
Show solution
Answer: C, $126°$. Convert $B$ to degrees: $-\frac{3\pi}{10} \cdot \frac{180}{\pi} = -54°$. $A$ and $B$ are supplementary, so $A = 180° - |-54°| = 180° - 54° = 126°$. $A$ is above the x-axis, so $A$ is positive.
Hard
A wheel of radius $8$ cm rotates through an angle of $k$ radians, sweeping an arc of length $12\pi$ cm. What is the value of $k$?
- A$\frac{12\pi}{8} \times \frac{180}{\pi} = 270$ (degrees, not radians)
- B$\frac{3}{2}$
- C$12\pi$
- D$\frac{3\pi}{2}$
Show solution
Answer: D, $\frac{3\pi}{2}$. Arc length $= r\theta$. So $12\pi = 8k$, giving $k = \frac{12\pi}{8} = \frac{3\pi}{2}$.
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