Radians to Degrees Formula

Radian to Degree Measure

The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.  In radians, one complete counterclockwise revolution is 2π2π and in degrees, one complete counterclockwise revolution is 360. So, the degree measure and radian measure are related by the equations.

A radian is a measurement of angle equal to the start to the end of an arc divided by the radius of the circle or arc. 1 radian is equal to 180/π, or about 57.29578°. There are about 6.28318 radians in a circle. The radian is the SI-derived unit for angle in the metric system.

Derivation of Radian to Degrees Formula

The conversion of the measure of an angle from radians to degrees can be done using the following formula: Angle in Radians × 180°/π = Angle in Degrees. For example, consider an angle π/9 rad. Now, using the radians to degrees formula, we have π/9 rad × 180°/π = (Angle in Degrees). Radians to degrees is a form of conversion used to convert the measurement of angles in geometry. To measure an angle, there are two different measuring systems. The two units used to measure an angle are radians and degrees. The unit radians is used mostly in the concept of trigonometry. The measure of angles can be converted from radians to degrees using a formula. To understand this formula and conversion of radians to degrees, we should understand the meaning of each unit of angle.

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How Do I Convert Degrees into Radians?

There are two different units used to measure an angle: radians and degrees. Hence, it is important for us to be proficient in the conversion of units of angle, that is radians to degrees and degrees to radians. When we take the radius of a circle and revolve it, we start constructing an angle that can be measured in radians or degrees.

From the above analysis, it is observed that one complete revolution of an angle at the center of a circle in radians is equal to 2 pi. Similarly, in degrees, one complete revolution of an angle made at the center of a circle is 360 degrees. Now if we compare both the values, we get,

360 degree = 2 * 3.142

1 degree = 2*3.142/360

1 degree = 3.142/180

Thus, the formula we use to convert degrees to radians:

Radians = Degrees * 3.142/180

This relation is deduced by analyzing a circle and can be used wherever you want to convert degrees into radians or radians into degrees.

Therefore, it is assumed that Radians = Degrees * 3.14/180 is the formula to convert degrees to radians in terms of pi.

Radian to Degrees Conversion Table

When we rotate the radius completely around the circle, it completes one rotation. The angle subtended at the center of the circle by the radius after one complete rotation is 2π radians. The angle in radians subtended by the radius at the center of the circle is the ratio of the length of the arc to the length of the radius. When the length of the arc becomes equal to the length of the radius, the angle subtended at the center becomes 1 radian. We denote the unit radian as rad. Radians is the SI unit of measuring angles.

1Rad × 180/π = 57.296 Deg

Radians to Degrees Chart

360°=2π360°=2π radians and

180°=π180°=π radians

From the latter, we obtain the equation 11 radian = (180π).  This leads us to the rule to convert radian measure to degree measure.  To convert from radians to degrees, multiply the radians by 180°π radians180°π radians.

Radian to Degrees Examples

To convert from degrees to radians, multiply the degrees by π180° radians. Angles are measured in degrees. One revolution is divided into 360 equal parts and each part is called a degree. The angle subtended at the center of the circle after one complete rotation of the radius is 360°. The symbol for degrees is denoted by radians. Degrees is not an SI unit to measure angles but it is an accepted unit to measure. Hence, while solving problems, it is preferred to convert the unit of angle from radians to degrees to understand it better. The instrument used to measure an angle in degrees is a protractor.

Comparing the measures of the angle for a complete rotation, we observe,

• 360 Degrees = 2π Radians
• 180 Degrees = π Radians

Radian to Degrees Conversion Of Some Common Angles  

The radians to degrees formula are used to convert radians to degrees. To convert radians to degrees we need to multiply the radians by 180°/π radians. When we measure angles, we use two types of units: degrees and radians, 1 degree is written as 1°. And 1 radian is written as 1 (or) 1c i.e., if there is no unit after the measure of an angle, it means that it is in radians.

1° = 0.017453 radians and 1 rad = 57.2958°. To convert an angle from degrees to radians, we multiply it by π/180°. To convert an angle from radians to degrees, we multiply it by 180°/π. One should essentially use radians when they are dealing with either object moving in circular paths or parts of a circular path. The problem statement can have the angle measured in degrees, but we should always convert the angles in degrees to radians before using them in any calculations. The length of the arc subtended by the central angle becomes the radian measure of the angle. This keeps all the important numbers like the sine and cosine of the central angle, on the same scale.

Example

Are the two lines in the diagram parallel if angle 3 = 35 and angle 5 = 150?

Since 35 + 150 = 185, the two lines are not parallel. This does not mean that angle 3 and angle 5 are not consecutive interior angles, it just means that the lines forming the angles are not parallel.