Degrees vs Radians
6 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
Degrees and radians are two ways to measure angles. Degrees split a full turn into 360 parts. Radians measure the angle by the length of arc it cuts on a circle of radius 1, so a full turn is 2π radians.
Key conversions
| Degrees | 0° | 30° | 45° | 60° | 90° | 180° | 360° |
|---|---|---|---|---|---|---|---|
| Radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 2π |
To convert degrees to radians multiply by π/180. To convert radians to degrees multiply by 180/π. One radian is about 57.2958°.
Why radians exist
Radians make formulas simple. The arc length of a circle is s = rθ only when θ is in radians. Derivatives of sin and cos are clean (d/dx sin x = cos x) only in radians. That is why calculus and physics use them.
The DEG/RAD switch on your calculator
The same keys give different answers depending on the mode. sin(30) is 0.5 in DEG but about −0.988 in RAD. sin(π/6) is 0.5 in RAD. If a trig answer looks strange, check the mode. In our scientific calculator the mode button sits at the top left of the display and is remembered next time you visit.
Worked conversions
- 135° × π/180 = 3π/4 ≈ 2.356 rad.
- 225° = 5π/4 ≈ 3.927 rad.
- 2 rad × 180/π ≈ 114.59°.
- π/5 rad = 180 ÷ 5 = 36°.
Exact values worth remembering
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° (π/6) | 1/2 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° (π/4) | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° (π/3) | √3/2 ≈ 0.866 | 1/2 | √3 ≈ 1.732 |
Arc length and sector area
With the angle θ in radians: arc length s = rθ and sector area A = ½r²θ. For r = 5 and θ = 2 rad, s = 10 and A = 25. The same formulas with degrees need an extra factor of π/180, which is exactly why radians are preferred in higher maths.
Which should I use?
- Degrees: school geometry, navigation, construction, everyday angles.
- Radians: calculus, physics, engineering and programming (most languages use radians).
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- A right triangle has legs 9 and 12. Find the hypotenuse.
Show answer
Hypotenuse c: 15
- Area of a circle with radius 7.
Show answer
Area: 153.93804
- Find log₂(64).
Show answer
log base 2 of 64: 6
Frequently asked questions
How many degrees is one radian?
About 57.2958°, which is 180 ÷ π.
How many radians are in 90 degrees?
π/2, about 1.5708.
Why does my calculator give a strange trig answer?
It is likely in the wrong angle mode.
Is a radian bigger than a degree?
Yes, one radian is about 57 times bigger than one degree.
Try it yourself
Pythagorean Theorem CalculatorFind the hypotenuse or a missing leg of a right triangle, plus area, perimeter and angles.Geometry CalculatorArea, perimeter, circumference, volume and surface area of common shapes.
Logarithm CalculatorCalculate logs in any base with change-of-base steps.Keep learning
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