Trigonometric Functions

How do you convert degrees to radians?

Quick answer

Multiply the angle in degrees by pi divided by 180 degrees (pi / 180) to convert the measurement into radians. Next, simplify the fraction to express the angle in exact terms with pi, or calculate the product for a decimal value. For example, 60 degrees times (pi / 180) equals pi / 3 radians.

The topic explained

Why we measure angles in radians

A degree is an arbitrary unit that divides a circle into 360 slices. A radian links an angle directly to the circle's radius. One radian is the measure of an angle formed at the center of a circle whose arc length equals the radius.

Because a full circle has a circumference of 2 * pi * r, one full rotation equals 2 * pi radians, or 360 degrees. Dividing both sides by 2 gives the foundational identity: 180 degrees equals pi radians. This ratio provides the exact conversion factor you use to change degrees into radians.

The conversion formula and steps

To switch from degrees to radians, multiply your degree value by pi / 180. Treat pi as an exact symbol rather than 3.14159, and reduce the fraction of degrees over 180 to lowest terms.

For example, convert 135 degrees to radians: multiply 135 by (pi / 180). Write it as (135 * pi) / 180. Divide numerator and denominator by 45 to get 3 * pi / 4 radians. To convert back from radians to degrees, reverse the process by multiplying by 180 / pi.

Placing angles on the unit circle

The unit circle has a radius of 1 centered at the origin (0, 0). Standard position starts at the positive x-axis (0 radians). Measure counterclockwise for positive angles and clockwise for negative angles.

Angles greater than 2 * pi radians (360 degrees) or negative angles wrap around the circle. To find their position, add or subtract full rotations of 2 * pi (or 360 degrees) to find a coterminal angle between 0 and 2 * pi. For instance, -pi / 6 radians lands at the same spot as 11 * pi / 6 radians.

Arc length and sector area formulas

Measuring angles in radians makes geometric formulas much simpler. When theta is in radians, the length of an arc is s = r * theta, where r is the radius.

The area of a sector of a circle is A = (1/2) * r^2 * theta. If r = 6 cm and theta = pi / 3 radians, the arc length is 6 * (pi / 3) = 2 * pi cm (about 6.28 cm), and the area is (1/2) * 36 * (pi / 3) = 6 * pi square cm.

Free printable worksheet

Degrees, Radians, and the Unit Circle Practice

Designed for 11th grade pre-calculus and trigonometry students. Solve each problem showing exact values in terms of pi unless asked for a rounded decimal.

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Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Degrees, Radians, and the Unit Circle PracticeName:

Convert 45 degrees to radians in simplest terms with pi.
Convert 210 degrees to radians in simplest terms with pi.
Convert 5 * pi / 3 radians to degrees.
Find a positive coterminal angle between 0 and 2 * pi for the angle -pi / 4.
Find a coterminal angle between 0 and 360 degrees for 420 degrees.
A circle has a radius of 10 cm. Find the arc length intercepted by a central angle of 2 * pi / 5 radians.
A pizza has a radius of 8 inches. What is the area of a slice with a central angle of pi / 4 radians?
Convert -120 degrees to radians and identify which quadrant its terminal side lies in.

Common questions

Why do mathematicians prefer radians over degrees?

Radians are a natural, dimension-free unit based on circle geometry rather than the arbitrary 360 slices of a circle. Formulas in calculus, physics, and advanced trigonometry become far simpler when angles are measured in radians.

What is the common mistake when converting degrees to radians?

Students often multiply by 180 / pi instead of pi / 180, flipping the conversion factor. Remember that units cancel out: to remove degree signs, the degrees (180) must be in the denominator.

Can an angle have a value greater than 2 pi?

Yes. An angle greater than 2 * pi represents more than one complete turn around the circle. For example, 9 * pi / 4 radians is one full revolution (2 * pi) plus an extra pi / 4 radians.

Do I have to keep pi in the final answer?

In high school math, exact form using pi is almost always preferred unless a question asks you to approximate a decimal value for a real-world measurement.

What topic comes next after radians and the unit circle?

Next, you will define sine and cosine as the y- and x-coordinates of coordinates on the unit circle, followed by graphing trigonometric functions like sine, cosine, and tangent.

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