Angles (Grade 5)
An angle is formed when two rays share a common starting point. Angles are all around us — in the corners of rooms, the hands of a clock, the opening of a door, and the edges of shapes.
In Class 5, you will learn to classify angles by their size, measure them using a protractor, and solve problems involving different types of angles. Understanding angles is essential for geometry, construction, navigation, and even sports.
Think of angles as turns. When you open a door, the door turns around the hinge. The amount of turn is the angle. A small opening is a small angle; a wide opening is a large angle.
What is Angles - Class 5 Maths (Geometry)?
An angle is formed by two rays (called the arms of the angle) that share a common endpoint called the vertex.
Angles are measured in degrees, using the symbol °. A full turn is 360°, a half turn is 180°, and a quarter turn is 90°.
Naming an angle: An angle can be named using three letters — the middle letter is always the vertex. For example, in angle ABC (written ∠ABC), B is the vertex, and BA and BC are the arms.
Key terms:
- Arms: The two rays that form the angle.
- Vertex: The common endpoint of the two rays.
- Interior: The region between the two arms of the angle.
- Exterior: The region outside the two arms of the angle.
Angles as turns: Imagine standing and facing North. If you turn to face East, you have turned 90° (a quarter turn). If you turn to face South, you have turned 180° (a half turn). Facing West is 270° (three-quarter turn). Facing North again is 360° (full turn).
Angles (Grade 5) Formula
Right Angle = 90° | Straight Angle = 180° | Complete Angle = 360°
Types and Properties
| Type | Measure | Example |
|---|---|---|
| Acute Angle | Greater than 0° and less than 90° | 30°, 45°, 60°, 89° |
| Right Angle | Exactly 90° | Corner of a book |
| Obtuse Angle | Greater than 90° and less than 180° | 120°, 135°, 150° |
| Straight Angle | Exactly 180° | A straight line |
| Reflex Angle | Greater than 180° and less than 360° | 210°, 270°, 350° |
| Complete Angle | Exactly 360° | Full rotation |
How to measure an angle with a protractor:
- Place the centre of the protractor on the vertex of the angle.
- Align the baseline of the protractor with one arm of the angle.
- Read the degree mark where the other arm crosses the scale.
- Use the inner scale if the arm opens to the left, and the outer scale if it opens to the right.
Solved Examples
Example 1: Example 1: Classifying Angles
Problem: Classify the following angles: 45°, 90°, 135°, 180°, 250°
Solution:
Step 1: 45° is between 0° and 90° → Acute angle
Step 2: 90° is exactly 90° → Right angle
Step 3: 135° is between 90° and 180° → Obtuse angle
Step 4: 180° is exactly 180° → Straight angle
Step 5: 250° is between 180° and 360° → Reflex angle
Answer: 45° (acute), 90° (right), 135° (obtuse), 180° (straight), 250° (reflex)
Example 2: Example 2: Clock Angles at 3 o'clock
Problem: What type of angle do the hands of a clock make at 3 o'clock?
Solution:
Step 1: At 3 o'clock, the minute hand points at 12 and the hour hand points at 3.
Step 2: The clock face is a full circle = 360°. It has 12 hours, so each hour = 360° ÷ 12 = 30°.
Step 3: From 12 to 3 = 3 hours = 3 × 30° = 90°.
Answer: The hands form a right angle (90°).
Example 3: Example 3: Clock Angle at 6 o'clock
Problem: What angle do the clock hands make at 6 o'clock?
Solution:
Step 1: At 6 o'clock, the minute hand is at 12 and the hour hand is at 6.
Step 2: From 12 to 6 = 6 hours = 6 × 30° = 180°.
Answer: The hands form a straight angle (180°).
Example 4: Example 4: Naming Angles
Problem: In triangle PQR, name all the angles.
Solution:
Step 1: A triangle has 3 vertices, so it has 3 angles.
Step 2: The angles are named by their vertices:
- ∠PQR (or ∠Q) — vertex at Q
- ∠QRP (or ∠R) — vertex at R
- ∠RPQ (or ∠P) — vertex at P
Answer: The three angles are ∠P, ∠Q, and ∠R.
Example 5: Example 5: Finding an Unknown Angle on a Line
Problem: Two angles on a straight line are 110° and x°. Find x.
Solution:
Step 1: Angles on a straight line add up to 180°.
Step 2: 110° + x° = 180°
Step 3: x° = 180° − 110° = 70°
Answer: x = 70°
Example 6: Example 6: Angles in a Right Angle
Problem: Two angles together make a right angle. One angle is 35°. Find the other.
Solution:
Step 1: A right angle = 90°.
Step 2: 35° + other angle = 90°
Step 3: Other angle = 90° − 35° = 55°
Answer: The other angle is 55°.
Example 7: Example 7: Reflex Angle Calculation
Problem: The non-reflex angle between two lines is 110°. What is the reflex angle?
Solution:
Step 1: A reflex angle and its corresponding non-reflex angle together make 360° (a complete turn).
Step 2: Reflex angle = 360° − 110° = 250°.
Answer: The reflex angle is 250°.
Example 8: Example 8: Angles in Real Life
Problem: Aditi opens a book flat on the table. What angle does the open book form?
Solution:
Step 1: When a book is fully opened flat, the two covers form a straight line.
Step 2: A straight line = 180°.
Answer: The open book forms a straight angle (180°).
Example 9: Example 9: Quarter, Half, and Full Turns
Problem: Express the following turns in degrees: (a) quarter turn, (b) half turn, (c) three-quarter turn, (d) full turn.
Solution:
(a) Quarter turn = 360° ÷ 4 = 90°
(b) Half turn = 360° ÷ 2 = 180°
(c) Three-quarter turn = 3 × 90° = 270°
(d) Full turn = 360°
Example 10: Example 10: Comparing Angles
Problem: Arrange in ascending order: 150°, 35°, 90°, 200°, 75°
Solution:
Step 1: Compare all the values.
Step 2: Ascending order (smallest to largest): 35° < 75° < 90° < 150° < 200°
Answer: 35°, 75°, 90°, 150°, 200°
Real-World Applications
Where do we use angles?
- Architecture: Builders use right angles to make walls straight and corners square. A wall not at 90° to the floor would lean and eventually fall.
- Sports: A cricket ball hit at different angles travels in different directions. Bowlers use angles to spin the ball. A basketball player calculates the angle of their throw to make a basket.
- Navigation: Pilots and sailors use angles to set direction. A compass rose shows 360° of directions.
- Clock reading: The angle between clock hands tells the time. Each minute = 6° on the clock face.
- Art and design: Rangoli patterns, mehndi designs, and tile patterns use angles for symmetry and repetition.
- Ramps and slopes: Wheelchair ramps are built at gentle angles. Steeper ramps have larger angles.
- Scissors and tools: When you open scissors, the two blades form an angle at the pivot.
Key Points to Remember
- An angle is formed when two rays meet at a common point (vertex).
- Angles are measured in degrees (°).
- Acute: 0° < angle < 90°. Right: exactly 90°. Obtuse: 90° < angle < 180°.
- Straight: exactly 180°. Reflex: 180° < angle < 360°. Complete: exactly 360°.
- Angles on a straight line add up to 180°.
- Angles at a point add up to 360°.
- Use a protractor to measure angles accurately.
- Each hour on a clock = 30°.
Practice Problems
- Classify these angles: 72°, 180°, 15°, 305°, 90°, 118°.
- What angle do the hands of a clock show at 9 o'clock?
- Two angles on a straight line are 65° and x°. Find x.
- What is the reflex angle if the smaller angle between two lines is 80°?
- Name the type of angle formed when you open a door halfway (about 90°).
- Kavi turns from facing North to facing East. What angle did he turn through?
- The three angles of a triangle are 50°, 60°, and x°. Find x. (Hint: angles of a triangle add up to 180°.)
- Draw an angle of 120° using a protractor and classify it.
Frequently Asked Questions
Q1. What is an angle in maths?
An angle is the figure formed when two rays share a common endpoint called the vertex. It is measured in degrees and describes the amount of turn between the two rays.
Q2. How many types of angles are there?
There are six types: acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), straight (exactly 180°), reflex (between 180° and 360°), and complete (exactly 360°).
Q3. How do you measure an angle with a protractor?
Place the centre of the protractor on the vertex. Align the baseline with one arm. Read the degree where the other arm crosses the scale. Use the correct scale (inner or outer) based on the direction of opening.
Q4. What angle do clock hands make at 12 o'clock?
At 12 o'clock, both hands overlap, so the angle between them is 0°.
Q5. What is the difference between a right angle and a straight angle?
A right angle is 90° (a quarter turn), like the corner of a square. A straight angle is 180° (a half turn), forming a straight line.
Q6. What is a reflex angle?
A reflex angle is greater than 180° but less than 360°. It is the larger angle on the other side when two lines meet. For example, if two lines form a 60° angle, the reflex angle is 360° − 60° = 300°.
Q7. Do angles on a straight line always add up to 180 degrees?
Yes. When angles are formed on one side of a straight line at a single point, their sum is always 180°. This is called the linear pair property.
Q8. What is a complete angle?
A complete angle is 360°. It represents one full rotation back to the starting position.
Q9. How do I find the angle between clock hands at any hour?
Each hour on the clock represents 30° (since 360° ÷ 12 = 30°). Multiply the number of hour spaces between the two hands by 30° to get the angle.
Q10. Is this topic in the NCERT Class 5 Maths syllabus?
Yes. Angles, their types, and measurement with a protractor are covered in the Geometry chapter of the NCERT/CBSE Class 5 curriculum.
Related Topics
- Parallel and Perpendicular Lines
- Angle Sum Property of Triangle
- Lines, Line Segments and Rays
- Types of Triangles (Grade 5)
- Quadrilaterals (Grade 5)
- Circles (Grade 5)
- Symmetry (Grade 5)
- Reflection and Rotation
- Nets of 3D Shapes (Grade 5)
- Views of 3D Shapes (Grade 5)
- Complementary and Supplementary Angles
- Constructing Triangles
