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Degrees to Radians: Definition and Examples | EDU.COM

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Degrees to Radians: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Degrees to RadiansDegrees to Radians: Definition and ExamplesTable of ContentsConverting Degrees to Radians: Formulas, Steps and Examples Definition of Degrees and Radians

A degree (°) is a unit for measuring the magnitude of an angle. It is the SI unit to measure angles. The measure of an angle in degrees is determined by the amount of rotation from the initial side to the terminal side. One complete counterclockwise rotation is 360∘360^{\circ}360∘. If it is divided into 360360360 equal parts, each part equals one degree. Thus, 111 degree equals 1360\frac{1}{360}3601​ of a complete revolution in magnitude.

Radian is another SI unit used to measure angles. One radian is the angle where the arc length equals the radius. For any circle, if radius = arc length, the angle is 111 radian. One radian is approximately equal to 57.29657.29657.296 degrees. The relationship between degrees and radians is given as 2π=360∘2\pi = 360^{\circ}2π=360∘ or π=180∘\pi = 180^{\circ}π=180∘. When we divide a circle in radians, about 3.143.143.14 radians will fit in each half of the circle. So, 6.286.286.28 radians will fit in a full circle. The exact amount of radians that fit in half a circle is πππ, which is about 3.143.143.14 radians.

Examples of Converting Degrees to Radians Example 1: Converting Positive Angle Measurements Problem:

Convert each degree measure into radians.

a) 120∘120^{\circ}120∘ b) 150∘150^{\circ}150∘ Step-by-step solution:

Step 1, Recall the formula to convert degrees to radians. The formula is: Angle in Radians = Angle in Degrees ×π180∘\times \frac{\pi}{180^{\circ}}×180∘π​

Step 2, For part a, substitute 120∘120^{\circ}120∘ into the formula.

120∘120^{\circ}120∘ angle in radians = 120∘×π180∘120^{\circ} \times\frac{\pi}{180^{\circ}}120∘×180∘π​

Step 3, Simplify the fraction.

120∘×π180∘=120π180=2π3120^{\circ} \times\frac{\pi}{180^{\circ}} = \frac{120\pi}{180} = \frac{2\pi}{3}120∘×180∘π​=180120π​=32π​

Step 4, For part b, substitute 150∘150^{\circ}150∘ into the formula.

150∘150^{\circ}150∘ angle in radians = 150∘×π180∘150^{\circ} \times\frac{\pi}{180^{\circ}}150∘×180∘π​

Step 5, Simplify the fraction.

150∘×π180∘=150π180=5π6150^{\circ} \times\frac{\pi}{180^{\circ}} = \frac{150\pi}{180} = \frac{5\pi}{6}150∘×180∘π​=180150π​=65π​ Example 2: Converting Radians to Degrees Problem:

Convert 2π6\frac{2\pi}{6}62π​ radians to degrees.

Step-by-step solution:

Step 1, Recall the formula to convert radians to degrees. The formula is: Angle in degrees = Angle in Radians ×180∘π\times \frac{180^{\circ}}{\pi}×π180∘​

Step 2, Substitute 2π6\frac{2\pi}{6}62π​ into the formula.

Angle in degrees = 2π6×180∘π\frac{2\pi}{6} \times \frac{180^{\circ}}{\pi}62π​×π180∘​

Step 3, Simplify the expression.

2π6×180∘π=2×180∘6=360∘6=60∘\frac{2\pi}{6} \times \frac{180^{\circ}}{\pi} = \frac{2 \times 180^{\circ}}{6} = \frac{360^{\circ}}{6} = 60^{\circ}62π​×π180∘​=62×180∘​=6360∘​=60∘ Example 3: Converting Negative Degrees to Radians Problem:

Convert −100∘-100^{\circ}−100∘ to radians.

Step-by-step solution:

Step 1, Remember that converting negative degrees works the same way as positive degrees. We use the same formula: Angle in radians = π180∘×\frac{\pi}{180^{\circ}} \times180∘π​× (angle in degrees)

Step 2, Substitute −100∘-100^{\circ}−100∘ into the formula.

Angle in radians = π180∘×(−100∘)\frac{\pi}{180^{\circ}} \times (-100^{\circ})180∘π​×(−100∘)

Step 3, Multiply to find the answer.

Angle in radians = π180∘×(−100∘)=−100π180\frac{\pi}{180^{\circ}} \times (-100^{\circ}) = -\frac{100\pi}{180}180∘π​×(−100∘)=−180100π​

Step 4, Simplify the fraction.

−100π180=−5π9-\frac{100\pi}{180} = -\frac{5\pi}{9}−180100π​=−95π​ radians Comments(3)HHistoryTutorEthanNovember 4, 2025This glossary page on degrees to radians has been a great help! It's made explaining the concept to my students so much easier.

MMasonFinnNovember 4, 2025This glossary page on degrees to radians is great! It's helped my students grasp the concept easily with its clear examples. Thanks!

MCMs. CarterSeptember 17, 2025I used the Degrees to Radians page to help my son with his math homework, and it made such a difference! The examples were super clear, and he finally got the hang of conversions.

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