In this article, we will discuss units of measuring angles and how to convert one unit into another.

There are three units to measure angles. These three units are revolutions, degrees, and radians which are discussed below in detail.

Revolutions

Revolution is one of the measure of angle. It forms when the initial side completes a full circle around its vertex until the side approaches its initial position. In other words, the initial and the terminal positions end up exactly in the same position. The revolution can be written as "rev". 1 revolution is equal to 360 ^ o.

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Degrees

This is the most common way of measuring angles. In a single revolution, there are 360 degrees. We can divide the degrees further, for instance, one degree is equal to 60 minutes and one minute is equal to 60 seconds. Hence, the angle that measures one second is equal to \frac {1}{3600} degrees. When we refer to the perpendicularity, we often mean an angle of 90 degree. We employ degrees to explain the triangles, like 15-75-90, 60-60-60, and 45-45-90. We write degrees with a tiny superscript circle at the top right side of a number. For instance, 90 degrees is written as 90^o.

Radian

We define radian geometrically and we denote it by the symbol rad. Previously, radian was a SI supplementary unit, however now it is a SI derived unit. Mathematicians often use radians as it is based on the radius of the circle. The center of the circle is the vertex of the angle. One radian is equal to the 180 degree divided by \pi. We can write it mathematically as:

1 rad = \frac {180 ^o} {\pi}

\frac {180 ^o} {\pi} is approximately equal to 57.2958 degrees which is represented as:

1 rad = 57.2958 ^ o

In the next section, we will solve some examples in which we will convert one unit of angle to another.

Example 1

Convert 180^o into radians.

Solution

We have already discussed that one radian is equal to 180 degrees divided by \pi. So, we can write it mathematically as:

1 rad = \frac {180 ^ o} {\pi}

\pi is in the denominator on the right hand side of the equation, so we will move it to the left hand side and multiply it with radians like this:

\pi rad = 180 ^ o

 

Example 2

Convert 360 ^o into radians.

Solution

One radian is equal to 180 degrees divided by \pi. So, we can write it mathematically as:

1 rad = \frac {180 ^ o} {\pi}

To isolate 180^o on the right hand side of the equation, we will move \pi on the left hand side like this:

\pi rad = 180 ^ o

We have to convert 360 ^ o into radians, so we will multiply the above equation by 2 like this:

2 \cdot \pi rad = 2 \cdot 180 ^ o

2 \pirad = 360 ^ o

 

Example 3

Convert 30 degrees into radians

Solution

One radian is equal to 180 degrees divided by \pi. So, we can write it mathematically as:

1 rad = \frac {180 ^ o} {\pi}

In this example, we don't have to convert 180 degrees to radians. Instead, we need to convert 30 degrees into radians. From the above formula, we can derive that 180 ^ o is equal to \pi rad.

180 ^ o = \pi rad

Similarly, one degree is equal to \frac {\pi}{180 ^o}:

1 ^ o = \frac {\pi}{180 ^o}

The angle \alpha in radians is equal to the the angle \alpha in degrees multiplied by \frac {\pi}{180 ^o}. Mathematically, we can write it as:

Radians = \frac {\pi} {180^o} \times degrees

= \frac {\pi} {180 ^o} \times 30 ^o

= \frac {\pi}{6} rad

Hence, 30 ^ o is equal to \frac {\pi}{6} rad

 

Example 4

Convert 2 rads into degrees.

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

To determine the value of 2 radians, we will multiply the expression by 2:

2 rads = \frac {180 ^ o} {\pi} \cdot 2

2 rads = (\frac {360} {\pi}) ^ o

 

Example 5

Convert 8 revolutions into radians

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

1 complete revolution is equal to 360 ^o. Hence, 8 revolutions will be equal to:

8 revolutions = 8 \cdot 360^o

8 \cdot 360 ^o = 16 \cdot 180 ^o = \pi rads

8 revolutions = 16 \pi rads

 

Example 6

Convert 4 revolutions into radians.

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

1 complete revolution is equal to 360 ^o. Hence, 4 revolutions will be equal to:

4 revolutions = 4 \cdot 360^o

4 \cdot 360 ^o = 8 \cdot 180 ^o = \pi rads

4 revolutions = 8 \pi rads

 

Example 7

Convert 5 \pi rads into degrees

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

 

To find the 5\pi rads, we will multiply both sides of the equation with 5 like this:

5 \cdot 180^o = 5 \pi rads

900 ^o = 5 \pi rads

 

Example 8

Convert \frac {2}{3} \pi rads into degrees

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

 

To find the \frac {2}{3} \pi rads, we will multiply both sides of the equation with \frac {2}{3} like this:

\frac {2}{3} \cdot 180^o = \frac {2}{3} \pi rads

120 ^o = \frac {2}{3} \pi rads

 

Example 9

Convert 4 rads into degrees.

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

To determine the value of 4 radians, we will multiply the expression by 4:

4 rads = \frac {180 ^ o} {\pi} \cdot 4

4 rads = (\frac {720} {\pi}) ^ o

 

Example 10

Convert 10 revolutions into radians

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

1 complete revolution is equal to 360 ^o. Hence, 10 revolutions will be equal to:

10 revolutions = 10 \cdot 360^o

10 \cdot 360 ^o = 20 \cdot 180 ^o = \pi rads

10 revolutions = 20 \pi rads

Example 11

Convert 9 \pi rads into degrees

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

To find the 9\pi rads, we will multiply both sides of the equation with 9 like this:

9 \cdot 180^o = 9 \pi rads

1200 ^o = 9 \pi rads

 

Example 12

Convert 11\pi rads into degrees

Solution

One radian is equal to 180 degrees divided by \pi.

1 rad = \frac {180 ^ o} {\pi}

Hence, 180^o is equal to \pi rads. We can write it as:

180 ^ o = \pirads

To find the 11\pi rads, we will multiply both sides of the equation with 11 like this:

11 \cdot 180^o = 11 \pi rads

1620 ^o = 11 \pi rads

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Rafia Shabbir