TOPIC 6 OF 41

Measuring and Drawing Angles with Protractor

Q: How do you measure an angle using a protractor?

To measure an angle using a protractor: place the centre of the protractor on the vertex of the angle, align the baseline with one arm of the angle, and read the degree mark where the other arm crosses the scale. Use the inner or outer scale depending on which arm is the baseline. This method is taught in NCERT Class 6 Chapter 2.

Q: How do you measure an angle using a protractor?

Place the centre point of the protractor exactly on the vertex of the angle. Align the baseline of the protractor along one arm of the angle. Read the degree value where the other arm intersects the protractor scale. Use the scale starting from zero on the aligned arm side. This is taught in NCERT Class 6 Chapter 2.

Q: How do you draw an angle of a given measure?

Draw a ray as one arm. Place the protractor centre on the endpoint with the baseline along the ray. Mark a point at the desired degree on the protractor scale. Remove the protractor and draw a ray from the endpoint through the marked point. The angle between the two rays is the required measure.

Q: What is the unit of angle measurement?

The standard unit of angle measurement is the degree, represented by the symbol. A full rotation equals 360 degrees, a straight angle is 180 degrees and a right angle is 90 degrees. The protractor typically shows markings from 0 to 180 degrees. NCERT Class 6 Maths uses degrees throughout Chapter 2.

Q: Why does a protractor have two scales?

A protractor has two scales (inner and outer) so that you can measure angles opening in either direction without repositioning the protractor. If you align one arm with the zero on the inner scale, read the other arm on the inner scale. This design makes angle measurement more convenient for students.

Q: What common mistakes occur when measuring angles?

Common mistakes include not placing the centre exactly on the vertex, not aligning the baseline properly with one arm, reading the wrong scale on the protractor, and parallax error from not looking straight down. NCERT Class 6 Chapter 2 warns students about these errors and provides practice.

🎓 Class 6 Mathematics CBSE Theory Ch 2 — Lines and Angles ⏱ ~35 min

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2.9 Measuring Angles

The Degree — A Unit to Measure Angles

To measure angles precisely, we need a standard unit. A degree^? (written as °) is the basic unit for measuring angles. One full turn (a complete circle) is divided into 360 equal parts, and each part is 1 degree.

📖 Definition

Degree (°): The unit for measuring angles. A full turn = 360°. A straight angle = 180°. A right angle = 90°.

🏛 Historical Context

The division of a circle into 360 parts goes back thousands of years. The Rigveda, one of the oldest texts of humanity, speaks of a wheel with 360 spokes. Many ancient Indian, Persian, and Babylonian calendars used 360-day years, linking the circle to the passage of time.

Key Angle Measures
Angle Type	Measure	Fraction of Full Turn
Full Turn	360°	1 (complete circle)
Straight Angle	180°	\(\frac{1}{2}\) turn
Right Angle	90°	\(\frac{1}{4}\) turn
Half of Right Angle	45°	\(\frac{1}{8}\) turn
One-third of Right Angle	30°	\(\frac{1}{12}\) turn
One Degree	1°	\(\frac{1}{360}\) turn

The Protractor

A protractor^? is the tool used to measure angles. It is a semicircle divided into 180 equal parts (degrees). It has two sets of markings — an inner scale (0° on the right) and an outer scale (0° on the left) — so you can measure angles opening in either direction.

How a Protractor Works

⚠️ How to Use a Protractor

Place the centre point of the protractor exactly on the vertex of the angle.
Align the base line (0° line) with one arm of the angle.
Read where the other arm crosses the scale.
Use the correct scale (inner or outer) — the one that starts at 0° on the arm you aligned.

Reading Angles on a Protractor

If one arm passes through 20° and the other passes through 55° on the same scale (say outer), then the angle between them is \(55° - 20° = 35°\). You can always find an angle by subtracting the smaller reading from the larger one.

Virtual Protractor

Bloom: L3 Apply

Set an angle with the slider and see how it would look on a protractor:

Angle: 45°

Making a Paper Protractor by Folding

You can make your own protractor using just a circular piece of paper and folding! Each fold bisects the angle, giving you precise angle measures.

Angles from Paper Folding
Number of Folds	Fraction of Circle	Angle Obtained
1 fold (half circle)	\(\frac{1}{2}\)	180°
2 folds (quarter)	\(\frac{1}{4}\)	90°
3 folds (eighth)	\(\frac{1}{8}\)	45°
4 folds (sixteenth)	\(\frac{1}{16}\)	22.5°

Figure it Out (Pages 35–44)

Figure for Q1 — Rays on a Protractor

Rays K (30°), W (50°), T (150°) from vertex A. Base ray is AL (0°).

Q1. Write the measures of the following angles: (a) \(\angle KAL\) (b) \(\angle WAL\) (c) \(\angle TAK\)

Answer:
(a) \(\angle KAL = 30°\) — Yes, it is possible to count the number of units in 5s or 10s on the protractor.
(b) \(\angle WAL = 50°\)
(c) \(\angle TAK = 120°\)

Fig. 2.20 — Straight Angle Divided into 8 Equal Parts

180° ÷ 8 = 22.5° per angle

Q2. In Fig. 2.20, \(\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = \_\_\_\_\). Why?

Answer: Each angle = 22.5°.
The straight angle of 180° is divided into 8 equal parts, so each angle = \(\frac{180°}{8} = 22.5°\).

Figure for Q3 — Protractor with Rays A, B, C, E at Vertex X

A at 0°, B at 45°, C at 75°, E at 180° (inner scale from right)

Q3. Find the degree measures of \(\angle BXE\), \(\angle CXE\), \(\angle AXB\), and \(\angle BXC\) from the protractor diagram above.

Answer: From the protractor figure (A at 0°, B at 45°, C at 75°, E at 180°):
\(\angle AXB = 45° - 0° = \mathbf{45°}\)
\(\angle BXC = 75° - 45° = \mathbf{30°}\)
\(\angle CXE = 180° - 75° = \mathbf{105°}\)
\(\angle BXE = 180° - 45° = \mathbf{135°}\)

Q4. Angles in a clock: (a) At 1 o'clock, the angle between the hands is 30°. Why? (b) What will be the angle at 2 o'clock? 4 o'clock? 6 o'clock?

Answer:
(a) A clock face is a full circle (360°) divided into 12 hours. Each hour = \(\frac{360°}{12} = 30°\). At 1 o'clock, the minute hand is at 12 and the hour hand is at 1, which is 1 gap = 30°.
(b) At 2 o'clock: \(2 \times 30° = 60°\)
At 4 o'clock: \(4 \times 30° = 120°\)
At 6 o'clock: \(6 \times 30° = 180°\) (straight angle)

Triangles for Q5 — Measure All Three Angles

Q5. Measure all three angles of each triangle above and add them up. What do you observe?

Answer: The sum of the three angles of any triangle is always 180° (a straight angle). This is called the angle sum property of a triangle. No matter what shape or size the triangle is, the three angles always add up to 180°.

Quadrilateral for Q6 — Measure All Four Angles

Q6. Measure all four angles of the quadrilateral ABCD above and add them up. What do you observe?

Answer: The sum of the four angles of any quadrilateral is always 360° (a full turn). A quadrilateral can be divided into two triangles, and \(2 \times 180° = 360°\).

2.10 Drawing Angles

To draw an angle of a given measure (say 30°), we use a protractor^? and follow a simple procedure.

⚠️ Steps to Draw an Angle

Draw the base: Draw a line segment (one arm of the angle). Label the starting point (vertex).
Place the protractor: Put the centre point on the vertex and align the base line with the arm.
Mark the angle: Find the required degree on the correct scale and make a dot.
Draw the arm: Remove the protractor and draw a ray from the vertex through the dot. This is the second arm.
Label: Write the angle measure.

Example: Drawing \(\angle TIN = 30°\)

\(\angle TIN = 30°\) — drawn using a protractor

Figure it Out (Pages 48–50)

Fig. 2.23 — Four Rays from Vertex O

List all possible angles: \(\angle AOB, \angle BOC, \angle COD, \angle AOC, \angle BOD, \angle AOD\)

Q1. In Fig. 2.23 above, list all the angles possible. Guess the measures of each, then measure with a protractor. How close were your guesses?

Answer: With 4 rays from a common point, the angles are: all pairs of adjacent and non-adjacent rays. For example: \(\angle AOB, \angle BOC, \angle COD, \angle AOC, \angle BOD, \angle AOD\). Measure each with a protractor and verify that your guesses improve with practice!

Q2. Use a protractor to draw angles of: (a) 110° (b) 40° (c) 75° (d) 112° (e) 134°

Answer: For each angle, follow the 5-step procedure:
1. Draw a base ray. 2. Place protractor centre on vertex. 3. Align base. 4. Mark the required degree. 5. Draw the second arm.
Verify by re-measuring — the measured value should match the intended value. These are all practical exercises to do in your notebook.

Let's Explore — Figure

\(\angle REB = 180°\) (straight), \(\angle RES = 90°\), \(\angle TER = 80°\)

Let's Explore: If \(\angle TER = 80°\), what is \(\angle BET\)? What is \(\angle SET\)? (Given: \(\angle REB\) is a straight angle, \(\angle RES = 90°\))

Answer:
From the figure, \(\angle REB = 180°\) (straight angle).
\(\angle BET = 180° - \angle TER = 180° - 80° = \mathbf{100°}\).

From the figure, \(\angle RES = 90°\) and T lies between R and S:
\(\angle SET = \angle RES - \angle RET = 90° - 80° = \mathbf{10°}\).

🔍 Activity 2.3 — Making a Paper Protractor

Bloom: L3 Apply

Materials needed: Circular piece of paper (trace a plate or use a compass)

🤔 PREDICT FIRST: If you fold a circular paper 4 times (each time in half), how many equal angles will you get when you unfold it? What will each angle measure?

Cut out a circle from paper
Fold 1: Fold in half — you get a semicircle. The fold gives 180°. Write "180°" at the bottom corner.
Fold 2: Fold in half again — you get a quarter circle. The new fold gives 90°. Write "90°".
Fold 3: Fold in half again — each part is 45°. Write "45°" and "135°".
Fold 4: Fold in half again — each part is 22.5°. Write all the new marks: 22.5°, 67.5°, 112.5°, 157.5°.
Unfold completely. You now have a paper protractor with markings at every 22.5°!

✅ Observation & Explanation

After 4 folds, you get \(2^4 = 16\) equal parts, each measuring \(\frac{360°}{16} = 22.5°\). The creases mark angles at 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5°, 180°, and continuing around the circle. This is essentially a protractor with 16 markings! By making more folds, you can get finer divisions.

📋

Competency-Based Questions

Scenario: A group of students are learning to use protractors for the first time. Their teacher gives them various angles to measure and draw. Some students make common mistakes while reading the scales.

Q1. A student reads an angle as 150° when it is actually 30°. What mistake did the student make?

L4 Analyse

(A) Used the wrong scale (inner instead of outer or vice versa)
(B) Placed the protractor upside down
(C) Did not align the centre point with the vertex
(D) Used a broken protractor

Answer: (A) — The student read from the wrong scale. The protractor has two scales that add up to 180° (30° on one scale corresponds to 150° on the other). Since the angle is clearly small (acute), the reading should be 30°, not 150°.

Q2. At what time(s) between 12:00 and 12:00 do the clock hands form exactly a 90° angle?

L3 Apply

Answer: The hands form a 90° angle at 3 o'clock (3 gaps = 3 × 30° = 90°) and at 9 o'clock (3 gaps going the other way). They also form 90° angles at other times during the day (approximately 12:16, 12:49, 1:22, 1:55, etc.) as the minute hand overtakes the hour hand repeatedly.

Q3. The Ashoka Chakra on the Indian flag has 24 spokes. What is the angle between two adjacent spokes? What is the largest acute angle formed between any two spokes?

L3 Apply

Answer:
Angle between adjacent spokes = \(\frac{360°}{24} = \mathbf{15°}\).
The largest acute angle is the largest multiple of 15° that is still less than 90°: \(5 \times 15° = 75°\). (6 spokes apart gives 90° which is a right angle, not acute.)
So the largest acute angle = 75°.

HOT Q. Puzzle: I am an acute angle. If you double my measure, you get an acute angle. If you triple my measure, you get an obtuse angle. What could I be?

L6 Create

Hint: Let the angle be \(x\). We need: \(x < 90°\) (acute), \(2x < 90°\) (double is acute), and \(90° < 3x < 180°\) (triple is obtuse).
From \(2x < 90°\): \(x < 45°\).
From \(3x > 90°\): \(x > 30°\).
So \(30° < x < 45°\). Any angle in this range works, e.g., \(x = 35°, 40°, 31°\), etc.

⚖️ Assertion–Reason Questions

Options:
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is NOT the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.

Assertion (A): The sum of the three angles of any triangle is 180°.

Reason (R): A straight angle measures 180°.

Answer: (B) — Both are true. The angle sum of a triangle is indeed 180°, and a straight angle is 180°. However, R does not directly explain A — the explanation involves the properties of parallel lines and transversals, not just the definition of a straight angle.

Assertion (A): Each hour on a clock face corresponds to a 30° angle.

Reason (R): A clock face is a circle of 360° divided into 12 equal parts.

Answer: (A) — Both are true. \(360° \div 12 = 30°\) per hour. R correctly explains why each hour corresponds to exactly 30°.

Frequently Asked Questions

How do you measure an angle using a protractor?

Place the centre point of the protractor exactly on the vertex of the angle. Align the baseline of the protractor along one arm of the angle. Read the degree value where the other arm intersects the protractor scale. Use the scale starting from zero on the aligned arm side. This is taught in NCERT Class 6 Chapter 2.

How do you draw an angle of a given measure?

Draw a ray as one arm. Place the protractor centre on the endpoint with the baseline along the ray. Mark a point at the desired degree on the protractor scale. Remove the protractor and draw a ray from the endpoint through the marked point. The angle between the two rays is the required measure.

What is the unit of angle measurement?

The standard unit of angle measurement is the degree, represented by the symbol. A full rotation equals 360 degrees, a straight angle is 180 degrees and a right angle is 90 degrees. The protractor typically shows markings from 0 to 180 degrees. NCERT Class 6 Maths uses degrees throughout Chapter 2.

Why does a protractor have two scales?

A protractor has two scales (inner and outer) so that you can measure angles opening in either direction without repositioning the protractor. If you align one arm with the zero on the inner scale, read the other arm on the inner scale. This design makes angle measurement more convenient for students.

What common mistakes occur when measuring angles?

Common mistakes include not placing the centre exactly on the vertex, not aligning the baseline properly with one arm, reading the wrong scale on the protractor, and parallax error from not looking straight down. NCERT Class 6 Chapter 2 warns students about these errors and provides practice.

Frequently Asked Questions — Lines and Angles

What is Measuring and Drawing Angles with Protractor in NCERT Class 6 Mathematics?

Measuring and Drawing Angles with Protractor is a key concept covered in NCERT Class 6 Mathematics, Chapter 2: Lines and Angles. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.

How do I solve problems on Measuring and Drawing Angles with Protractor step by step?

To solve problems on Measuring and Drawing Angles with Protractor, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 6 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

What are the most important formulas for Chapter 2: Lines and Angles?

The essential formulas of Chapter 2 (Lines and Angles) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.

Is Measuring and Drawing Angles with Protractor important for the Class 6 board exam?

Measuring and Drawing Angles with Protractor is part of the NCERT Class 6 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.

What mistakes should students avoid in Measuring and Drawing Angles with Protractor?

Common mistakes in Measuring and Drawing Angles with Protractor include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.

Where can I find more NCERT practice questions on Measuring and Drawing Angles with Protractor?

End-of-chapter NCERT exercises for Measuring and Drawing Angles with Protractor cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 2, and solve at least one previous-year board paper to consolidate your understanding.

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