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Measuring Angles
Measuring angles is done by using a simple geometric tool such as a protractor. A protractor is used more often than a compass. Measuring angles is useful in knowing at what measure the angle is exactly. In geometry, there are different types of angles that we encounter in our daily life and by using these tools we can find out the measurement of angles. Let us see how to measure an angle, the steps required, and solve a few examples.
1.  What is Measuring Angles? 
2.  Measurement of Angles Using Protractor 
3.  Constructing and Measuring Angles Using Protractor 
4.  Units of Measurement of Angles 
5.  FAQs on Measuring Angles 
What is Measuring Angles?
Measuring angles is done by using basic geometric tools like a protractor and a compass. These tools help in finding the exact measure of an angle. A protractor helps in providing the exact measure of the angle and a compass helps in constructing the angle. Measuring angles is done in three ways  degrees, radians, and revolution. Let us see what these three ways of measuring angles are.
Degrees
Degree is the unit of measure of an angle and is measured by using the geometric tool  a protractor. A degree is denoted by the symbol '°'. A circle completely rotates at a 360° and a degree is a part of that 360° rotation as it is divided into 360 equal parts. The different angles with different degrees are 30°, 45°, 90°, and so on. And is read as 30 degrees, 45 degrees, 90 degrees, etc.
Radians
Radian is another unit of measurement of an angle and is used in place of degrees when the angle needs to be measured in terms of radians. By definition, a radian is the ratio of the length of the arc that the angle subtends of a circle, divided by the length of the radius of the same circle. In other words, a radian is an angle subtended by the arc of the length of the radius of the same circle at the center and the ratio will give the radian measure of the angle. Radian is denoted as rad or ^{c} and is written as 1.7 radians or 1.7 rad or 1.7^{c}. Half a circle is 180° which is π radian and one complete revolution is 2π radian.
Revolution
Revolution is the simplest form of measuring angles. Along degrees, the revolution also is the unit of 360° as an angle is basically a subdivision of a circle rather than the sum of a few degrees. For example, while measuring in the revolution we can say a right angle is a quarter of a circle while in degrees an angle is read as a right angle is 90°.
Measurement of Angles Using Protractor
An angle is measured by using two geometric tools  a protractor and a compass. While a protractor can be used for both constructing and measuring, a compass is mostly used for constructing an angle. A protractor is considered one of the most important geometric tools as it helps in measuring angles in both degrees and radians. When we look at a protractor we can see measurements from 0 to 180 from left to right at the outer edge and 180 to 0 from right to left on the inner edge. The measurements in both the edges total up to 180°.
While measuring through a protractor the measure is usually in degrees. If the angle is on the left side of the protractor, we use the outer edge measurement and if the angle is on the right side of the protractor, we use the inner edge measurements. The steps to measure an angle are:
 Step 1: Place the center of the protractor on the vertex of the angle.
 Step 2: Superimpose one side of the angle with the zero line of the protractor.
 Step 3: The angle is equal to the number of degrees crossed on the protractor.
Let us look at an example. Measure ∠AOB.
Step 1: Align the protractor with the ray OB as shown below. Start reading from the 0° mark on the bottomright of the protractor.
Step 2: The number on the protractor that coincides with the second ray is the measure of the angle. Measure the angle using the number on the lower arc of the protractor.
Therefore, ∠AOB = 60°. Since the measure is greater than 0° and lesser than 90°, we can say that ∠AOB is an acute angle.
Example: Using the same example as above, let us measure the angle from the other side of the protractor. Measure ∠AOC.
Step 1: Align the protractor with the ray CO as shown below. Start reading from the 0° mark on the bottomleft of the protractor.
Step 2: The number on the protractor that coincides with the second ray is the measure of the angle. Measure the angle using the number on the top arc of the protractor.
Therefore, ∠AOC = 120°. Since the measure is greater than 90° and lesser than 180°, we can say that ∠AOC is an obtuse angle.
Constructing and Measuring Angles Using Protractor
A protractor can be used not only for measuring but also for constructing angles. This helps in both measuring the angles accurately and learning how to use the protractor. Let us see how to draw 40° using a protractor.
 Draw a baseline AB.
 Mark the point O and place the center of the protractor at O.
 Align the baseline of the protractor with the line OB.
 In the inner readings, look for 40º and mark it as point C.
 Now using a scale, join O and C.
 ∠COB = 40°
To measure 230º, a reflex angle, we can write it as 180° + 50°. We can just flip the protractor along the baseline and then mark 50°. 180° + 50° = 230°
Units of Measurement of Angles
The angles in geometry can be measured using mainly two units:
 Radians
 Degrees
Radians is the SI unit of measuring angles and is most commonly used in the concept of trigonometry. Another unit that is used for the measurement of angles is degrees, denoted by °, is commonly used in geometry. One revolution is equal to 360 degrees or 2π radians.
Important Notes on Measurement of Angles
 Measurement of angles is done by using basic geometric tools like a protractor and a compass.
 The angles in geometry can be measured using mainly two units  radians and degrees.
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Measuring Angles Examples

Example 1: In triangle ABC, use a protractor and measure ∠CAB.
Solution:Place the protractor baseline along the line AB and the center of the protractor at vertex A.
Observe the reading in the protractor which overlaps with line AC.Answer: Therefore, ∠CAB = 60°

Example 2: What is the angle formed at vertex B of the square ABCD?
Solution:
Place the protractor center at vertex B of the square and observe the reading.
Answer: ∠B = 90°

Example 3: Help Ben find the measurement of ∠ABC by looking at the image below.
Solution: By looking at the image, we can see that the ray AB is crossing the protractor at 153°. Therefore, ∠ABC = 153°.
FAQs on Measuring Angles
What is Meant by Measuring Angles?
In geometry, the process of reading angles or finding out the exact angle is called measuring angles. The best way to measure angles is by using the geometric tool known as the protractor. It is a semicircular instrument with markings from 0° to 180°.
What are the 3 Ways of Measurement of Angles?
In geometry, the 3 ways or units to measure angles are degree, radians, and revolution.
 Degree: The most common way to measure angles is degrees. A circle completely rotates at a 360° and a degree is a part of that 360° rotation as it is divided into 360 equal parts.
 Radians: A radian is an angle subtended by the arc of the length of the radius of the same circle at the center and the ratio will give the radian measure of the angle.
 Revolution: A revolution is the measure of the angle when it rotates one side completing a 360°.
What is the Best Way of Measuring Angles?
The best way measuring angles is by using a protractor. The steps are:
 Place the center of the protractor on the vertex of the angle.
 Superimpose one side of the angle with the zero line of the protractor.
 The angle is equal to the number of degrees crossed on the protractor.
How Do You Measure an Angle Without a Protractor?
To measure an angle without using a protractor, we use a simple ruler. We first connect the two rays at a point creating a triangle. Here are the steps:
 Draw a line connecting the two rays of the angle.
 Measure the length of the base of the triangle using a ruler. Also called as run.
 Measure the length of the straight side of the triangle using a ruler. Also called as rise.
 Once both the lengths are obtained we use a simple slope formula i.e. Slope = Rise/Run.
 Type the value for slope into a scientific calculator, then press the inverse tan button.
What are the Different Measures of Angles?
The different measures or types of angles are:
 Acute Angle: Measure between 0° to 90°.
 Obtuse Angle: Measure between 90° to 180°
 Right Angle: Measure is exactly 90°.
 Straight Angle: Measure is exactly 180°.
 Reflex Angle: Measure between 180° to 360°.
 Complete or Full Angle: Measure is exactly 360°.
Which Instrument is Used for Measurement of Angles?
The most commonly used instrument that is used for the measurement of angles is the protractor. A protractor is considered to be an important measuring instrument that is in the shape of a semicircle.
What are the Units Used for Measuring Angles?
The angles in geometry can be measured using mainly two units  radians and degrees.
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