Converting Between Radians and Degrees on the Digital SAT

Quick Answer: To convert degrees to radians, multiply by $\frac{\pi}{180}$; to convert radians to degrees, multiply by $\frac{180}{\pi}$. Always double-check if your Desmos calculator is set to the correct mode (degrees or radians) before solving trigonometry problems!

mindmap
  root((Angle Measures))
    Degrees
      Circle is 360
      Multiply by pi/180
    Radians
      Circle is 2pi
      Multiply by 180/pi
    Common Angles
      90 deg equals pi/2
      180 deg equals pi
      360 deg equals 2pi

What Is Converting Between Radians and Degrees?

On the Digital SAT, angles can be measured in two different units: degrees and radians. A full circle is $360^\circ$, which is equivalent to $2\pi$ radians. Therefore, a half-circle is $180^\circ$, or $\pi$ radians. The College Board tests your ability to seamlessly switch between these two units, especially in questions involving circle sectors, arc lengths, and trigonometry.

Understanding this conversion is crucial for the 2026 Digital SAT format. You will frequently encounter radians when working with the unit circle or when solving problems related to /sat/math/special-right-triangles-30-60-90 and /sat/math/special-right-triangles-45-45-90. If a problem gives you an angle in radians but asks for an answer in degrees (or vice versa), you must know how to apply the conversion factor.

Fortunately, the built-in Desmos Calculator makes handling these conversions and evaluating trigonometric functions much easier, provided you know which mode you are in!

Step-by-Step Method

1. Step 1 — Identify the unit you are starting with. Degrees usually have a degree symbol ($^\circ$), while radians are often (but not always) expressed in terms of $\pi$.
2. Step 2 — Choose your conversion factor. To convert degrees to radians, use $\frac{\pi}{180}$. To convert radians to degrees, use $\frac{180}{\pi}$. A helpful trick: put the unit you want in the numerator.
3. Step 3 — Multiply your starting angle by the chosen conversion factor.
4. Step 4 — Simplify the resulting fraction to get your final answer.

Desmos Shortcut

When taking the Digital SAT, the built-in Desmos graphing calculator is a powerful tool. By default, Desmos is in Radian mode. If a question asks you to evaluate $\sin(90^\circ)$ but Desmos is in radian mode, you will get the wrong answer!

To fix this, click the wrench icon (Graph Settings) in the top right corner of the Desmos window. At the bottom, you can toggle between "Radians" and "Degrees". If you just need to do a quick manual conversion, you can simply type 150 * (pi/180) directly into a Desmos line, and it will give you the decimal equivalent. You can then click the fraction button next to the result to convert that decimal back into a clean fraction involving $\pi$.

Worked Example

Question: An angle in the $xy$-plane has a measure of $\frac{7\pi}{6}$ radians. What is the measure of this angle in degrees?

A) $150^\circ$
B) $210^\circ$
C) $240^\circ$
D) $330^\circ$

Solution:

We need to convert from radians to degrees. Since we want degrees, we multiply by the conversion factor that has degrees in the numerator: $\frac{180}{\pi}$.

$\frac{7\pi}{6} \times \frac{180}{\pi}$

Notice that the $\pi$ in the numerator and the $\pi$ in the denominator cancel each other out:

$\frac{7 \times 180}{6}$

Next, simplify the fraction. We can divide $180$ by $6$ to get $30$:

$7 \times 30 = 210$

The measure of the angle is $210^\circ$.

The correct answer is B.

Common Traps

1. Forgetting to convert units — Based on Lumist student data, 15% of errors in the Geometry & Trigonometry domain involve forgetting to convert between degrees and radians entirely. Always highlight or underline what unit the question is asking for in the final answer.

2. Using the wrong calculator mode — Many students plug degree values into trigonometric functions while Desmos is still in radian mode. Our data shows that Geometry & Trigonometry has the highest overall error rate (27%), and small calculator setup mistakes contribute heavily to this. Always click the Desmos wrench icon to verify your mode before typing in sin, cos, or tan!