An angle measures 225^circ. What is the measure of this angle in radians?5/44pi/55pi/85pi/49pi/4
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
An angle measures \(225^\circ\). What is the measure of this angle in radians?
- \(\frac{5}{4}\)
- \(\frac{4\pi}{5}\)
- \(\frac{5\pi}{8}\)
- \(\frac{5\pi}{4}\)
- \(\frac{9\pi}{4}\)
1. TRANSLATE the problem information
- Given information:
- An angle measuring 225 degrees
- Need to convert to radians
2. INFER the conversion approach
- To convert degrees to radians, multiply by the conversion factor \(\pi/180\)
- This comes from the relationship: \(180° = \pi\) radians
3. SIMPLIFY through the conversion calculation
- Set up: \(225° \times (\pi/180) = 225\pi/180\)
- Now we have a fraction that needs to be reduced to lowest terms
4. SIMPLIFY the fraction to lowest terms
- Find the GCD of 225 and 180:
- 225 = 9 × 25 = \(3^2 \times 5^2\)
- 180 = 4 × 45 = \(2^2 \times 3^2 \times 5\)
- GCD = \(3^2 \times 5 = 45\)
- Divide both parts by 45: \(225\pi/180 = 5\pi/4\)
Answer: \(5\pi/4\) (Choice D)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Using the wrong conversion factor (\(180/\pi\) instead of \(\pi/180\))
Students sometimes remember there's a relationship between 180 and π but get confused about which direction the conversion goes. They might calculate \(225 \times (180/\pi)\), which would give a very large number that doesn't match any of the answer choices. This leads to confusion and guessing.
Second Most Common Error:
Inadequate SIMPLIFY execution: Not fully reducing the fraction or making arithmetic errors
Students correctly set up \(225\pi/180\) but either stop there without simplifying, or make calculation mistakes when finding the GCD. For example, they might divide by a smaller common factor like 5 instead of the full GCD of 45, getting \(45\pi/36\) instead of \(5\pi/4\). This may lead them to select Choice C (\(5\pi/8\)) if they make further arithmetic errors.
The Bottom Line:
This problem tests whether students can correctly apply the degree-to-radian conversion formula and then execute the algebraic simplification accurately. The key insight is remembering that π radians equals 180 degrees, so the conversion factor is \(\pi/180\).