**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.

**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.

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

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.

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

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.

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 \(40^ \circ\)

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

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

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

- The symbol for degree is (\(^\circ\)). Always use it while writing the angles. Example \(75^\circ , 90^\circ\)
- 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.
- To reduce errors, align the baseline and vertex properly.

**Solved Examples**

Example 1 |

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

**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 \(\text{AC}\).

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

Example 2 |

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

**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?

**Solution: **

Place the protractor along one arm and measure the angle.

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

Example 4 |

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

**Solution: **

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

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

Example 5 |

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

**Solution: **

We can use a protractor and measure the given angle.

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.

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.

1. | What is the measure of \( \angle\text{AOB}\)? |

2. | What angle do the hour hand and the minute hand of this clock make in the image below? |

**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:

- IMO Sample Paper Class 1
- IMO Sample Paper Class 2
- IMO Sample Paper Class 3
- IMO Sample Paper Class 4
- IMO Sample Paper Class 5
- IMO Sample Paper Class 6
- IMO Sample Paper Class 7
- IMO Sample Paper Class 8
- IMO Sample Paper Class 9
- IMO Sample Paper Class 10

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:

- Draw a line \(\text{OA}\)
- Place the protractor on the vertex \(\text{O}\) and align it with the line \(\text{OA}\).
- Look for the measure to be drawn on the protractor and mark a point \(\text{B}\).
- Join \(\text{OB}\).
- \(\angle\text{AOB}\) 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:

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

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

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