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Degrees (Angle)
A degree is a unit used to represent the measurement of an angle. There are two commonly used units of measurement of angles which are radians and degrees. In the case of practical geometry, we always measure the angle in degrees. A degree is represented by ° (degree symbol). The measure of a complete angle in degrees is 360 degrees (also written as 360°) which is the measure of one full rotation.
In this article, we will discuss the unit of angle known as degrees, its symbol, and its definition. We will also explore the relationship between radians and degrees, and their conversion. We shall a few solve examples giving the measure of an angle in degrees to understand the concept better.
1.  What is Degrees? 
2.  Angle in Degrees 
3.  Relationship Between Degrees and Radians 
4.  Measuring Angle in Degrees 
5.  FAQs on Degrees 
What is Degrees?
A degree is a measuring unit for angles. It is not an SI unit as the SI unit for measuring angles is radians. Generally, in geometry, we measure angles in degrees using a protractor. A protractor is generally used in schools to to measure angles to solve various mathematical problems. Let us see how the symbol of degrees is used to denote the measure of an angle.
Degrees Symbol
We use a small circle (°) after a number to its top right corner as a symbol of degrees. Let us see a few examples to understand how the symbol of degrees is used:
 30 degrees = 30°
 45 degrees = 45°
 π/3 radians = 60 degrees = 60°
 π/2 radians = 90 degrees = 90°
Angle in Degrees
The unit of measure for an angle in mathematics is called a degree. The degree of an angle is measured by using a tool called a protractor. A complete circle rotates at 360° and angles can be measured at different angles showcasing different degrees such as 30°, 45°, 60°, and so on. One rotation is divided into 360 equal parts, and each part is called a degree. We denote a degree with a circle °. For example, 180° means 180 degrees. An angle whose measure is given in degrees is called an angle in degrees.
We can classify angles according to the number of degrees as follows:
 Right Angle  Measurement of the right angle is 90 degrees (90°).
 Obtuse Angle  Measurement of an obtuse angle is greater than 90° and lesser than 180°
 Acute Angle  Measurement of an obtuse angle is less than 90° and greater than 0°
 Straight Angle  The Measurement of straight angle is 180°
 Reflex Angle  Measurement of the reflex angle is greater than 180° and lesser than 360°.
 Complete Angle  The Measurement of straight angle is 360°
Relationship Between Degrees and Radians
Angles are measured not only in degrees but also in radians. Radian is made by wrapping a radius along the circle. One complete counterclockwise revolution, in radians, is equal to 2π. We can convert the degree to radians and radians to degrees by using the following two formulas:
 To convert radians to degrees the formula is [Degrees = Angle in Radians × 180° / π]. One Radian is approximately equal to 57.2958 degrees.
 To convert degrees to radians the formula is [Radians = Angle in Degrees × π / 180°].
Given below is a table showing equivalent radian values for respective angles in degrees:
Angle in Degrees  Angle in Radians 

30°  π/6 
45°  π/4 
60°  π/3 
90°  π/2 
180°  π 
360°  2π 
As we discussed in the previous section, one complete rotation is equal to 360 degrees or 2π radians. We have another unit of measuring angles that is called, 'turn'. One turn is equal to one complete rotation. So, we have 1 turn = 360° = 2π radians.
Measuring Angle in Degrees
The best tool to measure angles in degrees is a protractor. The curved edge of the protractor is divided into 180 equal parts.
There are two sets of numbers on a protractor:
 One in a clockwise direction
 Another in a counterclockwise direction
If you look closely, the protractor has degrees marked from 0° to 180° from left to right on the outer edge and 180° to 0° on the inside.
Internal reading and external reading supplement each other. i.e., they add up to form a 180° degree angle.
Look at the above image, if the measured angle is on the left side from the center of the protractor, we will focus on the external readings of the protractor. In this case, ∠POR lies on the left side therefore ∠POR = 80°. If the measured angle is on the right side of the center of the protractor, we will focus on the internal readings of the protractor. In this case, ∠QOR lies on the right side therefore, ∠QOR = 100°. Here we have another example of measuring angle in degrees. Let's try to measure the angle ∠AOB in the figure given below with the help of a protractor.
 Step 1: Hold the protractor in such a way that the midpoint of the protractor coincides with vertex O of the given figure. Align the protractor perfectly with the ray OB as shown below.
 Step 2: Start reading from the 0° mark on the bottomright of the protractor. Measure the angle using the internal readings on the lower arc of the protractor.
Therefore, ∠AOB = 37°.
Important Notes on Degrees
 A degree is a unit of measuring angles.
 We can convert angle in degrees to angle in radians and vice versa using the following formulas:
 Angle in Degrees = Angle in Radians × 180° / π
 Angle in Radians = Angle in Degrees × π / 180°
 One complete rotation is equal to 360 degrees.
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Examples on Degrees

Example 1: Find the total number of degrees in three straight angles?
Solution: One straight angle = 180 degrees.
Three straight angles = one and a half full rotations
= 360 + 180 = 540 degrees.
Another approach to finding the number of degrees in three straight angles is by taking the product of the number of degrees in a straight angle by 3, that is,
= 3 × 180° = 540°.
Hence, with both the methods answer is the same.
Answer: Three straight angles = 540 degrees.

Example 2: Emma and James are cutting papers with the help of scissors to build some designs. James observes that during the cutting process, scissors make an angle. He asked Emma what angle in degrees is formed between the two arms of the given pair of scissors?
Solution: Place the protractor along one arm and measure the angle in degrees.
The angle which is to be measured is at the left part of the protractor. Therefore, use the outer readings of the protractor. Therefore, Angle formed = 30°
Answer: The measure of the angle in degrees between the two arms of the scissors is 30 degrees.

Example 3: Help Jack to convert the 60degree angle into radians.
Solution: The formula for the conversion of degrees into radians is, 1 degree = π/180 rad. Now multiply 60 on both sides,
1 degree = π/180 rad
⇒ 60 × degree = π/180 × 60° rad
= π/3 rad
Answer: 60 degrees = π/3 rad
FAQs on Degrees
What is Degrees in Geometry?
A degree is a unit used to represent the measurement of an angle. While measuring any angle we use the degrees symbol to denote it. It is denoted by °. For example, one full rotation is 360 degrees or (360°). One degree is equivalent to π/180 radians.
What is 1 Degree in Radians?
The measure of 1 degree in radians is given by, 1 degree = π/180° radians = 0.0174533 radians
How do you Define a Degree?
A degree, usually indicated by ° (degree symbol), is a measure of the angle. Angles can be of different measures or degrees such as 30°, 90°, 55°, and so on. To measure the degree of an angle, we use a protractor.
What Tools are Used to Measure the Angle in Degrees?
There are 5 Types of Tools to Measure Angles:
 Protractors
 Angle Gauge
 Multiple Angle Measuring Ruler
 Try Square
 Sine Bar
How To Convert an Angle in Degrees to Radians?
We can convert the measure of an angle in degrees to radians using the formula: Angle in Radians = Angle in Degrees × π / 180°.
How to Find Reference Angle in Degrees?
We can find the reference angle in degrees using the following formula,
 If the given angle x is less than 180 degrees, then its reference angle is given by 180°  x.
 If the given angle x is greater than 180 degrees, then its reference angle is given by x  180°.
How Many Degrees Are in a Half Turn?
Halfturn means making a straight angle. The measurement of a straight angle in degrees is 180°. Therefore, in halfturn, there are 180 degrees.
How Many Degrees are in a Full Turn?
Fullturn means making a complete angle. The measurement of a complete angle is 360°. Therefore, in one full turn, there are 360 degrees.
What are the Different Degrees of Angles we see in Geometry?
There are different types of angles according to their degrees in geometry. Let us see what they are:
 Right Angle  Measurement of the right angle is 90 degrees (90°).
 Obtuse Angle  Measurement of an obtuse angle is greater than 90° and lesser than 180°
 Acute Angle  Measurement of an obtuse angle is less than 90° and greater than 0°
 Straight Angle  The Measurement of straight angle is 180°
 Reflex Angle  Measurement of the reflex angle is greater than 180° and lesser than 360°.
 Complete Angle  The Measurement of straight angle is 360°
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