While most people know to measure angles in degrees, they may not be as familiar with radians. Degrees are more commonly taught to children at a young age and are used in areas such as geography and architecture, whereas radians are predominately used in mathematics and physics and are introduced to teenagers as they begin to learn advanced mathematics.
One way to look at the difference between the two is the viewpoint of the angle. While degrees are measured from the viewpoint of a stationary object, radians measure angles through the distance traveled. It can be helpful to measure angles from one or the other, depending on what you are measuring and the reason.
Converting degrees to radians and vice versa can be helpful if the other angles you are working with are measured in the opposite unit of measurement.
Continue reading to find out the differences between degrees and radians and how you can easily convert between the two.
Degrees help to measure how large angles are. A full revolution can be split into 360 equal parts, each of which is a degree. This means that half a revolution is 180 degrees, while a quarter of that (which is a right angle) is 90 degrees.
Although it isn't an official unit of measurement used by The International System of Units (SI), degrees are commonly used around the world to measure angles. It is common to convert from radians to degrees as they are often deemed easier to understand.
The symbol for degrees is '°', which is written after the angle measurement. Degrees can be manually measured using a protractor.
There are various names for certain angle sizes. They are as follows:
- Right Angle - angle is 90 degrees (90°).
- Obtuse Angle - angle is greater than 90° and less than 180°
- Acute Angle - angle is less than 90° and greater than 0°
- Straight Angle - straight angle is 180°, equal to two right angles
- Reflex Angle - angle is greater than 180° and lesser than 360°
- Complete Angle - straight angle is 360°
The Ancient Egyptians' measurement system was based on units of 60 and measured circles as 360. This is why circles in the modern era are still measured as 360°. The measurement system can also be seen in other areas of life. For example, there are 60 seconds in a minute and 60 minutes in an hour.
Positive degrees are when an angle is measured from a clockwise direction, while a negative degree is when an angle is measured from a counter-clockwise direction. You can convert between the two by subtracting 360 from a positive angle to find the negative angle or adding 360 to convert the other way. For example, a 30° is the same size as a 330° angle.
A radian is a unit of measurement that is based upon the radius of a circle. When you have a circle, you can rotate the radius completely around it for a full rotation. The angle subtended at the circle's center is equal to 2π radians.
The angle in radians subtended by the radius at the circle's center is equal to the ratio between the circle's arc length and the radius length. The figures that the angle measures are written as rad. Radians are the official SI unit of measurement for angles.
A single radian is equal to the angle that was formed when the angle of the radian subtends from the center of the circle to produce an arc. The length of this arc is equal to the radius of the circle. Radians are useful for calculus as they enable integral and derivative identities to be displayed in simple terms.
One degree is equal to 0.01745329252 radians. Another way of writing this is 1° = π/180° = 0.005555556π = 0.01745329252 rad. Pi radians are equal to 180°, as shown by π rad = 180°.
You can convert from degrees to radians by using the formula: Angle in Radians = Angle in Degrees × π/180
π radian is equal to 180°. Therefore, to convert from an angle in the measurement of degrees to radians, the given value must be multiplied by π/180.
The first step is to write down the size of the angle in degrees. You must then multiply this figure by π/180 and simplify the result by canceling the common factors. This final figure is the converted rate of the given degree into radians. The following is an example of these steps:
- You have a 90° angle
- Angle in radian = Angle in degree x (π/180)
- = 90 x (π/180)
- = π/2
Therefore, 90° is equal to π/2.
The following degrees to radians chart shows the conversion rate between a variety of different angle sizes:
| Degrees (°) | Radians (rad) |
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | (2π)/3 |
| 135° | (3π)/4 |
| 150° | (5π)/6 |
| 180° | π |
| 270° | (3π)/2 |
| 360° | 2π |
One complete anticlockwise revolution around a circle is equal to 2π radians, which is the same value as 360°. The relationship between the two can help show the formula to convert from degrees to radians.
- 2π radians = 360°
- π radians = 360°/2
- π radians = 180°
- 1 radian = 180°/π
This shows that you can convert from degrees to radians by multiplying the angle in radians by 180°/π. You can use the following formula to convert from radians to degrees:
Angle in Radians × 180°/π = Angle in Degrees
You know if an angle has been measured in radians if it has been written as 1 or 1c. This means that if you can't see a unit of measurement after the digit, the angle has likely been measured in radians. However, a degree would be written as 1°. One complete clockwise rotation is equal to 360° and has 360 equal degrees.
The degrees unit of measurement is used for various things, such as directions, forces, and acceleration. Degrees are the default measurement for angles outside of mathematics in areas such as architecture and geography.
They are used throughout the world, which makes them a good unit of measurement for international collaboration and trade. Many people find it easier to work with integers than fractions and π, which is how radians are written.
Degrees are more commonly taught in schools than radians. It is fairly easy to measure small angles using a protractor or draw a triangle based on specific angle sizes. this makes degrees preferable to radians in terms of smaller measurements because one radian is relatively large. A single radian is equal to 180°, which is double the size of a right angle.
While degrees are almost solely taught to primary school children, radians are introduced to secondary school children when they start to learn about arcs in circles and calculating radius and circumferences.
Degrees are more commonly used as the unit of measurement for areas not directly relating to mathematics, whereas radians are the preferred unit for pure mathematics and physics. They have various benefits, such as being dimensionless. This means that they can be treated as numbers rather than having a symbol tacked onto the end.
Radians also give a good description of the size of an angle, whereas the system of degrees is very arbitrary. For example, there is no real reason that there are 360 degrees in a complete circle or 90 degrees in a right angle other than it being the system that was created thousands of years ago.
Using radians as the unit of measurement enables you to relate the linear measurement with the size of the angle. A unit circle is a full circle that has a radius of one unit. The length of the arc subtended from the central angle of the circle is the radian measure of the angle. This means that figures for the sine and cosine are kept to the same scale on the central angle. For example, when graphing y = sin(x), one unit in the x-direction, it's the same as one unit in the y-direction. However, if you try to graph in degrees, the vertical scale must be extended a lot further to see that the graph goes up and down.
Degrees are the more commonly used unit of measurement in day-to-day life, but they are not the official unit of measurement used by The International System of Units (SI). This title is instead held by radians, which are usually reserved for measuring angles in mathematics and physics.
A full revolution of a circle is 360°, which is equal to 2π when measuring angles in units of radians. A radian is based on the radius of a circle and measures the angle that is subtended from the center of the circle. The angle is equal to the ratio between the length of the radius and the length of the circle's arc.
You can convert to degrees by multiplying the angle by π/180. This is because π is equal to 180° because a full circle is equal to two pi radians (2π) or a 360-degree measure. To convert from radians to degrees, you can multiply the angle by 180/π. When writing the measurement in degrees, the figure should be accompanied by a '°' symbol, but radians do not need to be accompanied by a symbol.
