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Angles

NCERT Class 5 · Mathematics Based on NCERT Class 5 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concept of angles as turns, helping students understand how much something has rotated from its original position. It connects real-life movements like turning around, clock hands, and opening doors to the idea of angles. The big idea is that an angle is a measure of turn, and we can classify turns as fractions of a full turn (like 1/4 turn, 1/2 turn) and name the corresponding angles (right, acute, obtuse, straight). By the end, students should be able to identify, measure, and draw angles using simple tools and understand clockwise and anti-clockwise directions.
Key Concepts (12)
Full turn
A complete rotation where the object comes back to its initial position.
e.g. When Ashutosh made a complete circle by rotating at one spot on the ground, he made a full turn.
Half turn
A rotation that is exactly half of a full turn, making a half-moon shape.
e.g. Sahana making a half-moon shape by rotating halfway around is a half turn.
Quarter turn
A rotation that is one-fourth of a full turn.
e.g. Pragya turning the yellow straw by 1/4 of a full turn to make a right angle.
Right angle
An angle formed by a quarter (1/4) of a full turn.
e.g. The corner of a notebook or the angle between the hands of a clock at 3:00.
Acute angle
An angle that is less than a quarter turn (less than a right angle).
e.g. The angle made when the yellow straw is turned less than 1/4 of a full turn.
Obtuse angle
An angle that is more than a quarter turn but less than a half turn.
e.g. The angle made when the yellow straw is turned more than 1/4 but less than 1/2 of a full turn.
Straight angle
An angle formed by two quarter turns (1/2 of a full turn).
e.g. The angle made when the minute hand moves from 12 to 6 on a clock.
Clockwise movement
The direction in which the hands of a clock move.
e.g. Opening a tap usually involves turning it clockwise.
Anti-clockwise movement
The opposite direction to the movement of the hands of a clock.
e.g. Unscrewing a lid or loosening a nut often requires anti-clockwise movement.
1/8 turn
A rotation that is one-eighth of a full turn, half of a quarter turn.
e.g. When the straw is turned by 1/8 of a full turn using the angle measuring tool.
1/12 turn
A rotation that is one-twelfth of a full turn, half of a 1/6 turn.
e.g. When the straw is turned by 1/12 of a full turn after folding a 1/6 part into half.
3/4 turn
A rotation that is three-fourths of a full turn, more than a straight angle.
e.g. When the minute hand moves from 12 to 9 on a clock.
Formulas & Equations (5)
1 full turn = 360 degrees (implied, not explicitly in text)
Reads as: One full turn equals three hundred sixty degrees.
Variables: No variables; it is a constant relationship.
Use when: When converting between turns and degrees in advanced problems.
⚠ Students sometimes think a full turn is 180 degrees instead of 360.
1/4 turn = Right angle
Reads as: One quarter turn equals a right angle.
Variables: No variables; it is a direct equivalence.
Use when: When identifying or drawing a right angle from a turn.
⚠ Students confuse a right angle with a straight angle.
1/2 turn = Straight angle
Reads as: One half turn equals a straight angle.
Variables: No variables; it is a direct equivalence.
Use when: When identifying or drawing a straight angle from a turn.
⚠ Students think a straight angle is the same as a right angle.
1/8 turn = Half of a quarter turn
Reads as: One eighth turn equals half of a quarter turn.
Variables: No variables; it is a fractional relationship.
Use when: When measuring smaller angles using the angle measuring tool.
⚠ Students forget that 1/8 turn is half of 1/4 turn, not half of 1/2 turn.
1/12 turn = Half of a 1/6 turn
Reads as: One twelfth turn equals half of a one-sixth turn.
Variables: No variables; it is a fractional relationship.
Use when: When using the circle folded into 6 equal parts and further halving.
⚠ Students confuse 1/12 turn with 1/8 turn.
How It Works
Making a paper fan to show angles
  1. Take a rectangular paper and fold it every 2 cm as shown in the picture.
  2. Paste ice cream sticks on the folds to create a paper fan.
  3. Hold one side fixed and turn the other side to show different turns.
  4. Observe the angles formed: acute (less than 1/4 turn), right (1/4 turn), obtuse (between 1/4 and 1/2 turn), straight (1/2 turn).
💡 The fixed side is the reference; the turning side creates the angle.
Making an angle measuring tool from a circle
  1. Cut out a circle from tracing paper.
  2. Fold the circle to make 8 equal parts (or 6 equal parts for another tool).
  3. Attach a straw to the centre of the circle and mark the starting point.
  4. Turn the straw to show different fractions of a turn (1/8, 2/8, 1/4, etc.).
  5. Cut out the 1/8 part and 2/8 parts (or 1/6 and 2/6 parts) and paste them on thicker paper to use as measuring tools.
💡 The starting point must be clearly marked; the straw rotates relative to it.
Measuring angles using the angle measuring tool
  1. Place the tool so the fixed side aligns with one arm of the angle.
  2. Align the straw with the other arm of the angle.
  3. Count how many 1/8 or 1/6 turns the straw has moved from the starting point.
  4. Combine measures if needed (e.g., 1/4 + 1/8 turn = 3/8 turn).
  5. State whether the angle is acute, right, obtuse, or straight based on the turn.
💡 Always start from the same reference line; otherwise the measurement will be wrong.
Determining direction (clockwise/anti-clockwise) of turns
  1. Observe the initial position of the object.
  2. Note the direction of movement: if it matches the movement of clock hands, it is clockwise; if opposite, it is anti-clockwise.
  3. Apply to real-life objects: opening a tap (usually clockwise), unscrewing a lid (usually anti-clockwise).
  4. Use the table to find ending direction after given turns from a starting direction (e.g., North, two right angles clockwise ends at South).
💡 Clockwise is the direction the hands of a clock move; anti-clockwise is the opposite.
Finding turns of the minute hand on a clock
  1. Note the initial position of the minute hand.
  2. Calculate the fraction of the circle moved: 15 minutes = 1/4 turn, 30 minutes = 1/2 turn, 45 minutes = 3/4 turn.
  3. For a given fraction of a turn, find the minutes: 1/12 turn = 5 minutes, 1/6 turn = 10 minutes, 4/12 turn = 20 minutes.
  4. Draw the final position of the minute hand on the clock face.
💡 60 minutes on a clock = 1 full turn; so 1 minute = 1/60 of a turn, but common fractions are 15 min = 1/4, 30 min = 1/2, etc.
Exam Traps — Don't Make These Mistakes
⚠ Confusing acute and obtuse angles
✗ An acute angle is more than a right angle.
→ Acute angles are less than a quarter turn (right angle), not more. Obtuse angles are more than a quarter turn but less than a half turn.
✓ An acute angle is less than a right angle (less than 1/4 turn). An obtuse angle is more than a right angle but less than a straight angle (between 1/4 and 1/2 turn).
⚠ Mixing up half turn and straight angle
✗ A half turn is a right angle.
→ A half turn is two quarter turns, which forms a straight angle, not a right angle. A right angle is only one quarter turn.
✓ A half turn is a straight angle (1/2 turn). A quarter turn is a right angle (1/4 turn).
⚠ Incorrectly identifying clockwise vs anti-clockwise
✗ Unscrewing a lid is clockwise.
→ Unscrewing a lid usually requires turning it anti-clockwise (opposite to clock hands). Clockwise is for tightening.
✓ Unscrewing a lid is anti-clockwise movement. Opening a tap is usually clockwise movement.
⚠ Wrong calculation of minutes from a fraction of a turn
✗ When the minute hand has turned by 1/6 of a full turn, it has moved by 6 minutes.
→ 1/6 of 60 minutes is 10 minutes, not 6. Students often divide 60 by the denominator incorrectly.
✓ When the minute hand has turned by 1/6 of a full turn, it has moved by 10 minutes (60 ÷ 6 = 10).
⚠ Thinking a 3/4 turn is a straight angle
✗ A 3/4 turn is a straight angle.
→ A 3/4 turn is more than a straight angle (which is 1/2 turn). A 3/4 turn is three quarters of a full turn.
✓ A 3/4 turn is more than a straight angle. A straight angle is only 1/2 turn.
Compare & Contrast
Direction of turn
Clockwise movement
Anti-clockwise movement
Clockwise movement is the direction in which the hands of a clock move; anti-clockwise movement is the opposite direction.
Size relative to a right angle
Acute angle
Obtuse angle
An acute angle is less than a quarter turn (right angle), while an obtuse angle is more than a quarter turn but less than a half turn.
Fraction of a full turn
Right angle
Straight angle
A right angle is a 1/4 turn, while a straight angle is a 1/2 turn (two right angles).
Exam Focus
Marks pattern: 2 MCQs (1 mark each) on identifying angles or turns, 1 Short Answer (2-3 marks) on clock hand turns or direction, 1 Long Answer (4-5 marks) on measuring angles using a tool or drawing angles from given turns.
Most asked: Identifying acute, right, obtuse, and straight angles from diagrams · Calculating turns of the minute hand on a clock (e.g., 15 minutes = 1/4 turn) · Clockwise and anti-clockwise directions in real-life contexts · Using fractions of turns to describe angles (1/4, 1/2, 3/4, 1/8, etc.)
Don't skip: Drawing angles for given turns (e.g., 1/2 turn, 1/4 turn, 1/6 turn) using a given line; Measuring angles in shapes and stating whether they are acute, obtuse, or right; Completing a table of starting direction, turns, and ending direction (clockwise/anti-clockwise)
One-Line Revision (read aloud before the exam)
  1. An angle is a measure of turn from one position to another.
  2. A full turn brings you back to the starting position.
  3. A half turn is also called a straight angle.
  4. A quarter turn is also called a right angle.
  5. An acute angle is less than a quarter turn.
  6. An obtuse angle is more than a quarter turn but less than a half turn.
  7. Clockwise movement is the same direction as the hands of a clock.
  8. Anti-clockwise movement is opposite to the hands of a clock.
  9. The minute hand moves 1/4 turn in 15 minutes.
  10. The minute hand moves 1/2 turn in 30 minutes.
  11. The minute hand moves 3/4 turn in 45 minutes.
  12. 1/8 turn is half of a quarter turn.
  13. 1/12 turn is half of a 1/6 turn.
  14. Two quarter turns make a half turn (straight angle).
  15. Four quarter turns make a full turn.
  16. A 3/4 turn is more than a straight angle.

MCQs — 10 Questions with Answers

Q1. When the minute hand of a clock moves from 12 to 3, it makes a _______ turn.

  • A. full
  • B. half
  • C. quarter ✓
  • D. eighth

Answer: C — From 12 to 3 is 15 minutes, which is 1/4 of the 60-minute circle, so it is a quarter turn.

Q2. Which angle is formed by a turn that is less than a quarter turn?

  • A. right angle
  • B. acute angle ✓
  • C. obtuse angle
  • D. straight angle

Answer: B — An acute angle is defined as any turn less than a quarter turn, making it small and pointed.

Q3. If Reema makes 2 quarter turns in the same direction, she has made a _______ turn.

  • A. quarter
  • B. full
  • C. half ✓
  • D. eighth

Answer: C — 1/4 + 1/4 = 2/4 = 1/2, which is a half turn.

Q4. The direction in which clock hands move is called _______.

  • A. anti-clockwise
  • B. counter-clockwise
  • C. clockwise ✓
  • D. sideways

Answer: C — Clockwise is the standard term for the direction that clock hands move from top to right to bottom.

Q5. An obtuse angle is formed by a turn that is:

  • A. less than 1/4 turn
  • B. exactly 1/4 turn
  • C. more than 1/4 turn but less than 1/2 turn ✓
  • D. exactly 1/2 turn

Answer: C — An obtuse angle is specifically between a quarter turn and a half turn, making it wider than a right angle.

Q6. Looking at a door opening from closed position, the maximum turn it can make is:

  • A. 1/4 turn (makes a right angle)
  • B. 1/2 turn (makes a straight angle)
  • C. more than 1/4 turn but less than 1/2 turn ✓
  • D. 1/8 turn

Answer: C — A typical door opens more than 90 degrees but less than 180 degrees, forming an obtuse angle.

Q7. When the minute hand moves by 30 minutes, it has made a _______ turn of the circle.

  • A. 1/4
  • B. 1/3
  • C. 1/2 ✓
  • D. 1/6

Answer: C — 30 minutes is exactly half of 60 minutes, so the minute hand makes a 1/2 turn.

Q8. Which object can make a full turn?

  • A. a scissors
  • B. a tap ✓
  • C. a clothes clip
  • D. a file cover

Answer: B — A tap can turn continuously and make a full complete turn, unlike scissors or clips which have limited movement.

Q9. Two acute angles added together can form an:

  • A. obtuse angle ✓
  • B. acute angle
  • C. right angle
  • D. straight angle

Answer: A — Two small turns (acute angles) together can make a larger turn that is between 1/4 and 1/2, which is an obtuse angle.

Q10. The angle marking in the house diagram is a right angle. This means it is a turn of:

  • A. 1/8 of full turn
  • B. 1/6 of full turn
  • C. 1/4 of full turn ✓
  • D. 1/2 of full turn

Answer: C — A right angle is always formed by exactly a 1/4 turn, which is 90 degrees of a full circle.

Flashcards

What is a full turn?

A full turn is when something rotates completely and comes back to its starting position, like a minute hand on a clock after 60 minutes.

What is a quarter turn?

A quarter turn is 1/4 of a full turn, and it creates a right angle, like when a door opens 90 degrees from the wall.

What is a half turn?

A half turn is 1/2 of a full turn, and it creates a straight angle, like when you turn around completely to face the opposite direction.

What is an acute angle?

An acute angle is formed by a turn that is less than a quarter turn, and it looks like a small opening, like the angle between two fingers slightly apart.

What is an obtuse angle?

An obtuse angle is formed by a turn that is more than a quarter turn but less than a half turn, like the angle at the top of a mountain shape.

What does clockwise mean?

Clockwise is the direction that the hands of a clock move, from top to right to bottom to left.

What does anti-clockwise mean?

Anti-clockwise is the opposite direction to clockwise, moving from top to left to bottom to right.

How many minutes does a 1/4 turn of the minute hand equal?

A 1/4 turn of the minute hand equals 15 minutes because the minute hand completes a full turn in 60 minutes.

What do two quarter turns equal?

Two quarter turns equal one half turn, which makes a straight angle.

What tool can you make to measure angles?

You can make an angle measuring tool by cutting a circle from tracing paper, folding it into equal parts, and attaching a straw at the center.

Important Board Questions

What is a quarter turn, and what angle does it make? [1 mark]

A quarter turn is 1/4 of a full circle. It creates a right angle, which you see when a door opens 90 degrees from a wall.

If the minute hand of a clock moves from 12 to 6, how many turns has it made? Name the angle formed. [2 marks]

From 12 to 6 is 30 minutes, which is half of 60 minutes. A half turn creates a straight angle.

Explain with an example from your home how clockwise and anti-clockwise movements are different. Give one example of each. [3 marks]

Clockwise is the direction clock hands move. Anti-clockwise is the opposite. Examples: opening a tap is clockwise (usually), unscrewing a lid might be anti-clockwise. Choose real objects you see at home.

You have made an angle measuring tool by folding a circle into 8 equal parts. (a) Show how you would measure a 3/8 turn using your tool. (b) Is a 3/8 turn an acute angle or obtuse angle? (c) What would be the measure if you added 2/8 turn and 3/8 turn together? [5 marks]

For (a), count 3 parts on your folded circle. For (b), check if 3/8 is less than 1/4 (acute) or between 1/4 and 1/2 (obtuse). For (c), add the fractions: 2/8 + 3/8 = 5/8, then check what angle this makes.

True or False? (a) A half turn and two quarter turns are the same. (b) An obtuse angle is smaller than a right angle. Give reasons for your answers. [2 marks]

For (a), calculate: two quarter turns = 2 × (1/4) = 1/2, so they are the same. For (b), remember that obtuse angles are wider (larger) than right angles, not smaller.

Assertion-Reason Questions for Angles

Q1. Assertion (A):

A quarter turn makes a right angle.

Reason (R):

A quarter turn is 1/4 of a full circle.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true, and the reason correctly explains why a quarter turn creates a right angle. A quarter of a complete rotation is 90 degrees, which is the definition of a right angle.

Q2. Assertion (A):

Two half turns make a full turn.

Reason (R):

A half turn is the same as an acute angle.

Answer: C — A is true but R is false — The assertion is true (1/2 + 1/2 = 1 full turn), but the reason is false. A half turn creates a straight angle, not an acute angle.

Q3. Assertion (A):

Clock hands move in the clockwise direction.

Reason (R):

Clockwise is the direction opposite to how a tap opens.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true. Clock hands do move clockwise, and most taps in India also open by turning clockwise. However, the reason does not explain why clock hands move clockwise; it just gives another example of clockwise motion.

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Case-Based Questions (CBQ) for Angles

Case Study:

Arun is playing with a paper fan in his classroom. He holds one side of the fan steady and turns the other side. First, he makes a turn that is less than a quarter turn, and the two sides make a sharp angle. Then, he makes another turn where the two sides form a perfect corner, like the corner of his math notebook. Finally, he turns the fan so much that the two sides form a straight line.

Q1. What type of angle did Arun make with his first turn?

  • A. Right angle
  • B. Acute angle ✓
  • C. Obtuse angle
  • D. Straight angle

Answer: B

Q2. In his second turn, what fraction of a full turn did Arun make?

  • A. 1/8 turn
  • B. 1/4 turn ✓
  • C. 1/2 turn
  • D. 3/4 turn

Answer: B

Q3. When Arun makes his final turn, what is formed between the two sides of the fan?

  • A. An acute angle
  • B. A right angle
  • C. An obtuse angle
  • D. A straight angle ✓

Answer: D

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