Protractor

Table of Contents 


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The Geometry Box 

A geometry box consists of many tools like protractor, triangle protractor, compass etc.

We use a protractor to measure angles.

Architects and designers use a more precise protractor called an angle protractor tool which gives more accurate measures.

A compass helps to construct an angle.

Set squares, also known as triangle protractor, are used to draw parallel and perpendicular lines.

The divider is used to measure the length between two points.


Introduction to Protractor Tool

Protractors have been in use since the seventeenth century. It is used for navigation in the sea and land.

Later by eighteenth century, protractors were widely used in geometry and mathematics.

an example of a protractor which looks like a semi-circle

Protractor: Definition

The common protractor is a simple, measuring instrument that is typically semicircle in shape with markings from \(0 ^\circ \) to \(180^\circ \) on it.


What is a Protractor Tool?

Protractor in maths is considered an important measuring instrument in the geometry box.

The protractor tool helps us measure an angle in degrees.

A radian scale protractor measures an angle in radians.

If you observe closely, a protractor tool has degrees marked from \(0\) to \(180\) from left to right in the outer edge and from\(180\) to \(0\) in the inner part.

 Protractor showing degrees marked along its inner part and outer edge

The inner readings and outer reading supplements each other. i.e, they add up to  \(180^ \circ\)

60 degrees and 120 degrees measured using a protractor

If the angle to be measured is at the left part of the protractor, we need to use the outer reading of the protractor.

\(\angle \text{AOC} = 60^ \circ\)

If the angle to be measured is at the right part of the protractor, we need to use the inner readings of the protractor.

\(\angle \text{BOC} = 120^ \circ\)


How to Use a Protractor?

An angle is formed between two rays.

Let us use the protractor and measure the angle \(\angle \text{COB}\).

Align the baseline of the protractor with the line \(\text{AB}\).

The center circle of the protractor at the vertex \(\text{O}\).

Now, identify the reading on the protractor where Ray \(\text{C}\) is extending.

   Angle BOC is marked 30 degrees with a protractor

\[\angle \text{COB} = 30^\circ\]

What do you think is the measure of \(\!\angle \text{AOC}\)?

If the angle to be measured is rotated or tilted, tilt your protractor to align with the base ray \(\text{OA}\) and then measure.

 Measuring 45 degrees with the protractor

In the above image,

\(\angle \text{AOB}\ = 45^ \circ \) 


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Measuring Angles from Protractor

The usage of a protractor in maths is considered essential as it helps measure angles.

In the previous section, we learnt how to use a protractor.

Here is an interactive simulation for you to measure different angles.

  • Click on Create New Angle in the simulation given below.
  • A new angle \( \angle \text{AOB}\) will be generated.
  • Using the protractor, measure \( \angle \text{AOB}\) and enter the value in the space provided.
  • Click on Check to verify your answer.


Draw Angles with the Protractor

Let us see how to draw an angle of \(30 ^\circ \) using a protractor. 

  • Draw a baseline \(\text{OA}\)
  • Mark the point \(\text{O}\) and place the center of the protractor at \(\text{O}\).
  • Align the baseline of the protractor with the line \(\text{OA}\).
  • In the outer readings, look for \(30 ^\circ \) and mark it as point \(\text{B}\)
  • Now using a scale, join \(\text{O}\) and \(\text{B}\)

Look at the simulation below on how to draw a \(30 ^\circ \) angle.

To measure \(220^\circ \), a reflex angle, we can write it as \(180^\circ + 40^\circ\)

We can just flip the protractor along the baseline and then mark \(40^ \circ\)

220 degree angle measured using a protractor

\(180 ^\circ +40^\circ  = 220^ \circ\)

We can also use a \(360^\circ \) protractor to draw this angle. 

360 degree protractor tool

The angle protractor tool and \(360^\circ \) protractor gives more precise measurements.

 
important notes to remember
Important Notes
  1. The symbol for degree is (\(^\circ\)). Always use it while writing the angles. Example \(75^\circ , 90^\circ\)
  2. If the angle to be measured is in the left side, use the outer readings of the protractor and if the angle to be measured is in the right side, use the inner readings.
  3. To reduce errors, align the baseline and vertex properly. 


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Solved Examples

Example 1

 

 

In \( \Delta \text{ABC}\), use a protractor and measure \( \angle \text{CAB}\).

An example of a triangle ABC

Solution:

Place the protractor baseline along the line \(\text{AB}\) and the center of the protractor at the vertex \(\text{A}\).

Observe the reading in the protractor which overlaps with line \(\text{AC}\).

Measure of angle A of a triangle ABC

\(\therefore \angle \text{CAB} = 60^\circ\)
Example 2

 

 

In the given figure, what is the measure of \( \angle \text{AOC} \)?

an Angle BOC is marked as 30 degrees on a protractor

Solution:

The ray \(\text{OC}\) is pointing to \(150^\circ\)

\(\therefore \angle \text{AOC} = 150^\circ\) 
Example 3

 

 

What is the angle formed between the two arms of the given pair of scissors?

example to find the angle formed between the arms of the scissors

Solution:

Place the protractor along one arm and measure the angle.

Angle formed between the arms of the scissors as seen with a protractor is 30 degrees.

\(\therefore \text{Angle formed} = 30^\circ\) 
Example 4

 

 

What is the angle formed at vertex \(\text{B}\) of the square \( \text{ABCD}\)?

 

An example of a square ABCD

Solution:

Place the protractor center at vertex \(\text{B}\) of the square and observe the reading. 

Angle B of the square example measures 90 degrees

\(\therefore \angle \text{B} = 90^\circ\)  
Example 5

 

 

Use a protractor and verify if the angle beneath the staircase is \(140^\circ\).

Angle formed beneath the staircase

Solution:

We can use a protractor and measure the given angle.

Angle formed beneath the staircase is 150 degrees.

The angle formed is \(150^\circ\) and not \(140^\circ\)

Common Mistakes

1. The center of the protractor (marked in the image below) has to align exactly with the vertex of the angle to be measured. 

Even a little misalignment will result in an incorrect reading.

Center of the protractor

2. The baseline of the protractor has to align exactly over the line for which we are measuring the angle, else the degree measurement will be inaccurate.

A protractor placed on the baseline exactly along the angle to be measured.

 
Challenge your math skills
Challenging Questions
1. What is the measure of \( \angle\text{AOB}\)?
  An angle measuring 270 degrees
2.  What angle do the hour hand and the minute hand of this clock make in the image below? 
  Angle between the hour and minute hand of the clock

Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. How to use a protractor to draw angles?

To draw angles using a protractor, the following steps can be followed:

  1. Draw a line \(\text{OA}\)
  2. Place the protractor on the vertex \(\text{O}\) and align it with the line \(\text{OA}\).
  3. Look for the measure to be drawn on the protractor and mark a point  \(\text{B}\).
  4. Join \(\text{OB}\).
  5. \(\angle\text{AOB}\) is the required angle.

Refer Draw Angles with the Protractor section to know more.

2. How to use a 360 degree protractor?

You can use a 360 degree protractor by following these simple steps:

  1. Place the center of the \(360^ \circ \) protractor at the vertex.
  2. The \(0 ^\circ \) degree line should align with the baseline of our angle to be measured.
  3. Observe the number on the \(360^\circ \) protractor where the other ray is pointing. 

3. How do you use a protractor step by step?

  1. Align one arm of the angle to be measured with the baseline of the protractor.
  2. Align the vertex with the center of the protractor.
  3. Carefully observe the reading on the protractor where the other ray is pointing.
  
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