Lesson Explainer: Conversion between Radians and Degrees Mathematics

In this explainer, we will learn how to convert radians to degrees and vice versa.

Radians, like degrees, are a unit of measurement for angles. They are used as an alternative to degrees. Let us formally define what we mean by a radian.

We start with a circle of center ๐‘‚ and radius ๐‘Ÿ.

Now imagine that we take another length of the radius ๐‘Ÿ, which we curve along the circumference of the circle.

We now have a sector of the circle with radius ๐‘Ÿ and arc length ๐‘Ÿ. The angle at the center of the circle is defined as 1 radian (rad).

Definition: Radians

One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

Now that we have defined what we mean by one radian, we will demonstrate how to convert between radians and degrees.

We recall that the distance around a circle is its circumference, and its measure is found by multiplying ๐œ‹ by the diameter, ๐‘‘. We can write this circumference, ๐ถ, as ๐ถ=๐œ‹๐‘‘.

Equivalently, we can also describe the circumference in terms of the radius as ๐ถ=2๐œ‹๐‘Ÿ.

Looking at the sector of the circle above, with an arc length of ๐‘Ÿ, we consider how many arcs of length ๐‘Ÿ can be drawn on the circumference. As we defined the arc length, ๐‘Ÿ, to be the same as the radius, ๐‘Ÿ, then this means that the number of arcs that could fit on the circumference would be 2๐œ‹๐‘Ÿรท๐‘Ÿ=2๐œ‹.

If we sketch a diagram of this, we can recall that a numerical approximation for 2๐œ‹ is 6.283โ€ฆ. On the diagram, there are 6 complete arcs and a remaining portion of 0.283โ€ฆ arcs.

Since 2๐œ‹ arcs make up the full circumference of the circle, then there must be 2๐œ‹ sectors making up the circle itself. Since the angle of each sector is 1 radian, we can say that the sum of these angles is 2๐œ‹ radians.

However, we also know that the sum of the angles in a circle, in degrees, is 360โˆ˜, which means that we can observe that 2๐œ‹=360.radiansโˆ˜

This gives an ideal conversion between the two units of measurement. Additionally, we can also divide both sides of this equation by 2 or by 4, giving two alternative conversion facts: ๐œ‹=180radiansโˆ˜ and ๐œ‹2=90.radiansโˆ˜

Similarly, taking the equation ๐œ‹=180radiansโˆ˜, we can divide both sides by ๐œ‹ to find the value of 1 radian in degrees:1=๏€ผ180๐œ‹๏ˆ,1โ‰ˆ57.296.radianradian(to3decimalplaces)โˆ˜โˆ˜

Knowing any of these conversion facts will allow us to convert any angle from radians to degrees, or vice versa.

How To: Converting between Radians and Degrees

2๐œ‹=360๐œ‹=180๐œ‹2=90radiansradiansradiansโˆ˜โˆ˜โˆ˜

We will now look at some examples of how to convert angles in degrees to angles in radians.

Example 1: Converting an Angle from Degrees to Radians

Convert the following angle measures from degrees to radians. Give your answers in terms of ๐œ‹ in their simplest form.

  1. 90โˆ˜
  2. 30โˆ˜
  3. 55โˆ˜

Answer

In order to convert an angle in degrees into one in radians, we remember the conversion fact: 180=๐œ‹.โˆ˜radians

Part 1

In the first part of this question, we need to convert the angle of 90โˆ˜ to radians. We notice that 90โˆ˜ is half of 180โˆ˜. Therefore, we can divide both sides of the conversion by 2, which gives 180=๐œ‹,โˆ˜radians90=๐œ‹2.โˆ˜radians

We can, therefore, give the solution that 90โˆ˜ in radians is ๐œ‹2.radians

Note that, because we need to give our answer in terms of ๐œ‹, we leave our answer in this form, rather than converting to a decimal.

Part 2

To find the angle of 30โˆ˜ in radians, we take the equation 180=๐œ‹โˆ˜radians and divide both sides by 6, giving 30=๐œ‹6.โˆ˜radians

Alternatively, we could also use the angle measurement we worked out in the first part of the question, that is, 90=๐œ‹2,โˆ˜radians and divide both sides by 3, giving 30=๐œ‹2ร—3=๐œ‹6.โˆ˜radians

Using either method, we see that 30โˆ˜ in radians is ๐œ‹6.radians

Part 3

When we need to convert an angle in degrees that is not a factor of 180โˆ˜, it is often simplest to first find the measure of 1โˆ˜ in radians.

We know that 180=๐œ‹.โˆ˜radians

So, dividing both sides of this equation by 180 would give 1=๐œ‹180.โˆ˜radians

As we need to convert 55โˆ˜ to radians, we can multiply both sides of this equation by 55. This gives us 55=55ร—๏€ป๐œ‹180๏‡=55๐œ‹180.โˆ˜radiansradians

To simplify the fraction, we divide both the numerator and denominator by the highest common factor of 5, which gives 55=11๐œ‹36.โˆ˜radians

Therefore, 55โˆ˜ is equivalent to 11๐œ‹36 radians.

In the previous question, we saw how to convert the angle of 55โˆ˜ into radians, first by dividing by 180โˆ˜ and then by multiplying it by ๐œ‹. The same method can be applied to find any angle measurement.

How To: Converting an Angle from Degrees to Radians

To change any angle in degrees to one in radians, multiply the given angle by ๐œ‹180: ๐œƒ=๐œƒ๐œ‹180.โˆ˜radians

In the following question, we will consider how to change an angle given in radians into one given in degrees.

Example 2: Converting an Angle from Radians to Degrees

Convert ๐œ‹3 radians to degrees.

Answer

To convert an angle from radians to degrees, we remember that ๐œ‹=180.radiansโˆ˜

We notice that to convert ๐œ‹3 radians into degrees, we will divide both sides of this equation by 3, which gives ๐œ‹3=๏€ผ1803๏ˆ=60.radiansโˆ˜โˆ˜

Thus, ๐œ‹3 radians can be written in degrees as 60โˆ˜.

When we are converting angles that are simple factors of ๐œ‹ radians or 180โˆ˜, we can utilize the method we have seen in the previous questions. However, there is an efficient way to convert an angle from radians to degrees in one step.

How To: Converting an Angle from Radians to Degrees

To change any angle in radian measure into degrees, we multiply the given angle by 180๐œ‹: ๐œƒ=๏€ฝ180๐œƒ๐œ‹๏‰.radiansโˆ˜

We will now see how we can apply this in the following example.

Example 3: Converting an Angle from Radians to Degrees

Convert 0.5 rad to degrees giving the answer to the nearest second.

Answer

We recall that to change any angle in radian measure into degrees, we multiply the given angle by 180๐œ‹: ๐œƒ=๏€ฝ180๐œƒ๐œ‹๏‰.radiansโˆ˜

In this question, we can substitute the given angle, 0.5 radians, for the angle ๐œƒ. This gives 0.5=180ร—0.5๐œ‹=90๐œ‹=28.6788โ€ฆ.radiansโˆ˜โˆ˜โˆ˜

Next, we need to convert the angle of degrees in decimal form into one in degrees, minutes, and seconds. To do this, we take the whole part of the answer as degrees. So, 28.6788โ€ฆโˆ˜ gives 28โˆ˜.

For the minutes, we multiply the remaining decimal by 60, and use the whole part of the answer as the 38 minutes. This gives 0.6788โ€ฆร—60=38.873โ€ฆ=38.minutes

For the seconds, we multiply the new remaining decimal by 60 and round the answer to the nearest whole number. This gives us 0.873โ€ฆร—60=52.40โ€ฆ=52.seconds

Hence, we can give the answer that 0.5 rad is equivalent to 2838โ€ฒ52โ€ฒโ€ฒโˆ˜.

In the following example, we will demonstrate an application of these conversion formulas in a more complex example.

Example 4: Solving a Problem Involving Degrees and Radians

Find the value of the two angles in degrees given that their sum is 74โˆ˜ and their difference is ๐œ‹6 radians. Give your answer to the nearest degree.

Answer

We begin this question by defining the two unknown angles as ๐‘ฅ and ๐‘ฆ. We are given that their sum in degrees is 74โˆ˜ and their difference in radians is ๐œ‹6 radians. We can then form two different equations: ๐‘ฅ+๐‘ฆ=74,๐‘ฅโˆ’๐‘ฆ=๐œ‹6.โˆ˜radians

We can solve these equations for ๐‘ฅ and ๐‘ฆ if the angles are given in the same units of measurement. Therefore, we can use the conversion between radians and degrees: ๐œ‹=180.radiansโˆ˜

We can change either both angles to degrees, or both to radians, but as we are asked for the final answer in degrees, then it is most efficient to change the angle of ๐œ‹6 in radians to one in degrees.

Comparing ๐œ‹=180radiansโˆ˜ and ๐œ‹6, we notice that we must divide through by 6. Therefore, ๐œ‹=180๐œ‹6=๏€ผ1806๏ˆ=30.radiansradiansโˆ˜โˆ˜โˆ˜

Alternatively, we could use the rule whereby to change an angle in radians to degrees, we multiply the given angle by 180๐œ‹. So, to change ๐œ‹6 radians to degrees, we do the following: ๐œ‹6=๏€ผ๐œ‹6ร—180๐œ‹๏ˆ=๏€ผ1806๏ˆ=30.radiansโˆ˜โˆ˜โˆ˜

Either method demonstrates that ๐œ‹6=30radiansโˆ˜, which can then be substituted into the second equation, ๐‘ฅโˆ’๐‘ฆ=๐œ‹6, to give ๐‘ฅโˆ’๐‘ฆ=30.โˆ˜

We can now solve the pair of simultaneous equations using the substitution or elimination method:

๐‘ฅ+๐‘ฆ=74,โˆ˜(1)

๐‘ฅโˆ’๐‘ฆ=30.โˆ˜(2)

To solve by eliminating the ๐‘ฆ variable first, we add equations (1) and (2) as follows: ๐‘ฅ+๐‘ฆ=74+๐‘ฅโˆ’๐‘ฆ=302๐‘ฅ+0=104โˆ˜โˆ˜โˆ˜

To solve this equation for ๐‘ฅ, we divide both sides by 2, which gives ๐‘ฅ=52.โˆ˜

Next, substituting ๐‘ฅ=52โˆ˜ into the first equation, ๐‘ฅ+๐‘ฆ=74โˆ˜, and rearranging, gives 52+๐‘ฆ=74๐‘ฆ=74โˆ’52๐‘ฆ=22.โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜

Therefore, we have found that ๐‘ฅ=52โˆ˜ and ๐‘ฆ=22โˆ˜, so we can give the value of the two angles in degrees as 52,22โˆ˜โˆ˜.

In the following question, we can use our knowledge of converting between radians and degrees to help solve a problem involving the angles in a triangle.

Example 5: Solving a Problem Involving Degrees and Radians

Two angles in a triangle are 55โˆ˜ and 7๐œ‹18 radians. Find the value of the third angle giving the answer in radians in terms of ๐œ‹.

Answer

We can sketch a triangle with the two given angles of 55โˆ˜ and 7๐œ‹18 radians.

We will need to find the third angle in this triangle as an angle in radians, so we can define this angle to be ๐‘ฅ radians.

To convert the angle of 55โˆ˜ to radians, we multiply the angle by ๐œ‹180. This would give 55=55๏€ป๐œ‹180๏‡=55๐œ‹180=11๐œ‹36.โˆ˜radians

The three angles in the triangle can then be written as 11๐œ‹36 radians, 7๐œ‹18 radians, and the unknown angle, ๐‘ฅ radians.

To find the unknown angle, we can recall that the angles in a triangle sum to 180โˆ˜. However, as we are considering the angles in radians here, and as 180=๐œ‹โˆ˜radians, we can also say that, in radians, the angles in a triangle sum to ๐œ‹ radians

Using this information, we can form an equation in ๐‘ฅ: 11๐œ‹36+7๐œ‹18+๐‘ฅ=๐œ‹.

We can rewrite and simplify, giving 11๐œ‹36+14๐œ‹36+๐‘ฅ=๐œ‹25๐œ‹36+๐‘ฅ=๐œ‹.

Subtracting 25๐œ‹36 from both sides gives ๐‘ฅ=๐œ‹โˆ’25๐œ‹36=๐œ‹๏€ผ1โˆ’2536๏ˆ=๐œ‹๏€ผ3636โˆ’2536๏ˆ=11๐œ‹36.

The missing angle in the triangle is 11๐œ‹36radians.

We will now summarize the key points.

Key Points

  • Radians and degrees are both units of measurement of angles.
  • One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
  • We can convert between degrees and radians using
    • 2๐œ‹=360radiansโˆ˜,
    • ๐œ‹=180radiansโˆ˜,
    • ๐œ‹2=90radiansโˆ˜.
  • Alternatively, we can convert an angle in degrees to one in radians by multiplying the given angle by ๐œ‹180, and we can convert an angle in radians to one in degrees by multiplying the given angle by 180๐œ‹.

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