Radians and Degrees

Most people learn to measure angles in degrees: 90° for a right angle, 360° for a full turn. Calculus does something different. It measures angles in radians, and it does so for a very practical reason, which this lesson gets to below. The good news is that converting between the two takes a single multiplication, and after a little practice the common angles in radians, , , , , become as familiar as 30°, 45°, 90° and 180°.

What Is a Radian?

Draw a circle and mark off an arc exactly as long as its radius. The angle that arc makes at the centre is one radian.

Since a circle’s circumference is , exactly of those radius-length arcs fit around it. So a full turn is radians, and half a turn is radians:

That single fact is the basis of every conversion. It also tells you that one radian is , a bit less than a sixth of a turn.

How to Convert Radians to Degrees

Multiply by :

Example 1: An Angle With π

Convert radians to degrees.

The s cancel, which is what usually happens: most textbook angles are written as fractions of .

Example 2: A Plain Number of Radians

Convert 2.5 radians to degrees.

There’s no to cancel here, so the answer is a decimal. An angle written without , like 2.5, is still in radians unless it has a degree sign.

How to Convert Degrees to Radians

Multiply by , then simplify:

Example 3: Degrees to an Exact Radian Value

Convert to radians.

Divide 225 and 180 by their greatest common factor, 45, and leave in the answer. In calculus, the exact form is almost always preferred over the decimal 3.927.

Example 4: An Awkward Angle

Convert to radians.

Common Angles in Radians and Degrees

A quick way to rebuild the table: is and is ; every other entry is a multiple of one of those.

Negative Angles and Angles Bigger Than 2π

Angles aren’t limited to one turn. A positive angle turns counterclockwise from the positive -axis and a negative angle turns clockwise, so points straight down, the same direction as . Angles that end in the same place are called coterminal, and you find them by adding or subtracting full turns of (or ):

The conversion formulas work exactly the same way for these angles: is , and is . In calculus this matters because trig functions repeat every , so an angle and all its coterminal angles give the same sine and cosine.

Why Calculus Uses Radians

Radians aren’t just a different unit; they make the formulas of calculus simple. Two facts sit at the heart of the trig parts of the course:

Both are true only when is in radians. Measure in degrees and the limit becomes instead of 1, and every derivative of a trig function picks up an extra factor of . Radians are the unit in which those messy constants disappear, because a radian is defined by arc length, the same kind of length the rest of the problem is measured in.

There’s a practical warning in that, too: when you evaluate a trig function on a calculator during a calculus problem, check that it’s in RAD mode, not DEG.

Arc Length and Sector Area

Radians also give the simplest formulas for parts of a circle. With in radians and radius :

For a circle of radius 6 cm and , the arc is cm, and the sector’s area is cm². In degrees, both formulas would need an extra . The circles lesson covers the rest of the circle’s geometry.

Common Mistakes to Avoid

• Using the wrong fraction. To get degrees, multiply by , so the cancels. To get radians, multiply by , so the degrees cancel.
• Treating a plain number as degrees. means the sine of 2 radians, not 2 degrees.
• Leaving the calculator in degree mode for a calculus problem.
• Not simplifying. should be reduced to .
• Converting to 180 inside a radian answer. radians is 180°, but on its own is the number 3.14159…, not 180.

Practice Problems

Work each one before opening the answer.

Problem 1. Convert radians to degrees.

Problem 2. Convert to radians.

Problem 3. Convert 1.2 radians to degrees, to two decimal places.

Problem 4. Convert to radians in exact form.

Problem 5. Convert radians to degrees.

Problem 6. A circle has radius 8 cm. How long is the arc cut off by a central angle of ?

With radians in hand, the trigonometric functions themselves come next: what sine, cosine and tangent mean on the unit circle, and how their graphs behave.