Radians and Degrees SAT — SAT Math Guide
Radians and degrees SAT questions test your ability to convert between these two angle measurement systems and apply them in trigonometric contexts. Both units measure angles, but radians use the relationship between arc length and radius while degrees divide a circle into 360 equal parts. You'll encounter 2-3 questions involving angle conversions in the SAT math section, often combined with trigonometry or circle problems. With the right conversion formulas and practice, these problems become straightforward point-scorers.
What You Need to Know
One full rotation = 360° = 2π radians
π radians = 180 degrees (the key conversion relationship)
Common angles: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
Radians often appear as fractions of π (like π/4, 3π/2)
Calculator can work in both degree and radian modes
📐 KEY FORMULA: degrees = radians × (180/π) and radians = degrees × (π/180)
💡 PRO TIP: Memorize that π radians = 180°. Everything else follows from this relationship!
How to Solve Radians and Degrees Problems on the SAT
Example Question 1 — Medium Difficulty
An angle measures 5π/6 radians. What is this angle measure in degrees?
A) 120°
B) 130°
C) 150°
D) 160°
Solution:
Step 1: Use the conversion formula degrees = radians × (180/π)
Step 2: Substitute 5π/6 for radians: degrees = (5π/6) × (180/π)
Step 3: Simplify: degrees = (5π × 180)/(6π) = 900/6 = 150°
✅Answer: C — When converting from radians to degrees, multiply by 180/π and the π terms cancel out.
Example Question 2 — Hard Difficulty
A wheel rotates through an angle of 240°. If the wheel has a radius of 8 inches, what is the arc length traveled by a point on the rim, in terms of π?
A) 16π/3 inches
B) 32π/3 inches
C) 64π/3 inches
D) 128π/3 inches
Solution:
Step 1: Convert 240° to radians: 240° × (π/180) = 240π/180 = 4π/3 radians
Step 2: Use arc length formula: s = rθ where θ is in radians
Step 3: Calculate: s = 8 × (4π/3) = 32π/3 inches
✅Answer: B — Arc length problems require radians, so convert degrees first, then apply s = rθ.
Common SAT Math Mistakes to Avoid
❌Mistake: Forgetting to convert between radians and degrees when the problem requires it
✅Fix: Always check what unit the answer needs and convert accordingly
❌Mistake: Using degrees in formulas that require radians (like arc length)
✅Fix: Arc length and sector area formulas need radians, so convert first
❌Mistake: Writing π/180 instead of 180/π when converting radians to degrees
✅Fix: Remember "radians to degrees" means multiply by 180/π
❌Mistake: Leaving answers in decimal form when the question asks for exact values
✅Fix: Keep answers in terms of π when possible, especially for common angles
Practice Question — Try It Yourself
The central angle of a sector is π/3 radians. What is this angle in degrees, and what fraction of the complete circle does this sector represent?
A) 60°, 1/6 of the circle
B) 60°, 1/4 of the circle
C) 45°, 1/8 of the circle
D) 72°, 1/5 of the circle
Show Answer
Answer: A — π/3 × (180/π) = 60°. Since a full circle is 360°, this sector is 60°/360° = 1/6 of the circle.
Key Takeaways for the SAT
Master the fundamental conversion: π radians = 180 degrees
Always check whether your calculator is in degree or radian mode
SAT math radians and degrees problems often connect to arc length and sector area
Common angle measures appear frequently: memorize π/6 = 30°, π/4 = 45°, π/3 = 60°
When in doubt, convert to the unit system the formula requires
Related SAT Math Topics
Strengthen your SAT math prep with these related topics:
Arc length sector area →
Unit circle trigonometry →