What are the types of angles?

Acute ( 90° and 180° and < 360°).

What is a perpendicular line?

Two lines are perpendicular if they intersect at right angles (90°). Symbol: ⊥

What are the types of triangles based on sides?

Scalene (all sides different), Isosceles (two sides equal), Equilateral (all three sides equal).

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NCERT Solutions for Class 6 Maths Chapter 5 — Understanding Elementary Shapes

Q: What is a perpendicular line?

Two lines are perpendicular if they intersect at right angles (90°). Symbol: ⊥

Now that you know the basic ideas of geometry, it is time to measure and classify them. This chapter teaches you how to measure line segments and angles, identify perpendicular lines, classify triangles and quadrilaterals, recognise polygons, and describe three-dimensional shapes. Every solution below follows a step-by-step method so you see the reasoning, not just the answer.

Key Concepts

Concept	What It Means	Example
Measuring Line Segments	Finding the length of a line segment using a ruler or comparing with a divider. Align the ruler's zero mark with one endpoint and read the value at the other.	Segment AB = 5.3 cm
Types of Angles	Angles classified by their measure: Acute (< 90 degrees), Right (= 90 degrees), Obtuse (> 90 and < 180 degrees), Straight (= 180 degrees), Reflex (> 180 and < 360 degrees), Complete (= 360 degrees)	A right angle is formed by two perpendicular lines
Measuring Angles	Using a protractor to find the measure of an angle in degrees. Place the centre of the protractor on the vertex and align the base line with one ray.	Angle = 45 degrees
Perpendicular Lines	Two lines that intersect at a right angle (90 degrees). The symbol for perpendicular is an inverted T.	The corner of a notebook, where two edges meet at 90 degrees
Classification of Triangles (by sides)	Scalene (all sides different), Isosceles (two sides equal), Equilateral (all three sides equal)	Equilateral triangle: all sides = 5 cm
Classification of Triangles (by angles)	Acute-angled (all angles < 90 degrees), Right-angled (one angle = 90 degrees), Obtuse-angled (one angle > 90 degrees)	Right-angled triangle: one angle exactly 90 degrees
Quadrilaterals	A polygon with four sides. Types include rectangle, square, parallelogram, rhombus, trapezium. Each has 4 sides, 4 vertices, and 2 diagonals.	A rectangle has opposite sides equal and all angles 90 degrees
Polygons	Closed figures made of line segments. Named by number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8)	A stop sign is an octagon
Three-Dimensional Shapes	Shapes that have length, breadth, and height (depth). Examples: cube, cuboid, cylinder, cone, sphere, pyramid	A brick is a cuboid; a dice is a cube

Exercise 5.1 — Measuring Line Segments

Question 1: What is the disadvantage in comparing line segments by mere observation?

Solution

Step 1: When we compare line segments just by looking at them, our eyes can be tricked.
Step 2: Two line segments that look almost equal may actually differ by a small amount.
Step 3: Visual comparison is not precise or reliable.

AnswerComparing by mere observation can lead to errors. Line segments that appear equal may differ slightly. We should use a ruler or divider for accurate comparison.

Question 2: Why is it better to use a divider than a ruler, while measuring the length of a line segment?

Solution

Step 1: A ruler has thickness and markings that may cause positioning errors.
Step 2: The eye may not be directly above the ruler when reading (this is called parallax error).
Step 3: A divider has two sharp points that can be placed exactly on the two endpoints, and then the gap can be measured on a ruler.

AnswerA divider avoids positioning and parallax errors. It gives a more accurate measurement because the two sharp tips can be placed exactly on the endpoints.

Question 3: Measure the lengths of line segments AB and CD using a ruler. Which is longer?

Solution

Step 1: Place the ruler with its zero mark exactly at point A.
Step 2: Read the marking at point B. Suppose AB = 4.6 cm.
Step 3: Similarly, place the ruler at zero on point C and read the marking at point D. Suppose CD = 5.2 cm.
Step 4: Compare: 5.2 cm > 4.6 cm.

AnswerSegment CD is longer than segment AB.

Question 4: If A, B, C are three points on a line such that AB = 5 cm, BC = 3 cm, and AC = 8 cm, which point lies between the other two?

Solution

Step 1: Check if AB + BC = AC.
Step 2: 5 cm + 3 cm = 8 cm = AC. So the condition is satisfied.
Step 3: When AB + BC = AC, point B lies between A and C.

AnswerPoint B lies between A and C because AB + BC = AC.

Exercise 5.2 — Angles: Right, Straight, and Complete

Question 1: What fraction of a clockwise revolution does the hour hand of a clock turn through when it goes from: (a) 3 to 9 (b) 4 to 7 (c) 7 to 10 (d) 12 to 9 (e) 1 to 10?

Solution

A full clockwise revolution = 12 hours = 360 degrees. One hour = 360/12 = 30 degrees. A half revolution = 6 hours. A quarter revolution = 3 hours.

(a) 3 to 9:

Step 1: From 3 to 9, the hour hand moves 6 hours.
Step 2: 6 out of 12 hours = 6/12 = 1/2 of a revolution.

Answer1/2 (half revolution)

(b) 4 to 7:

Step 1: From 4 to 7, the hour hand moves 3 hours.
Step 2: 3 out of 12 hours = 3/12 = 1/4 of a revolution.

Answer1/4 (quarter revolution)

(c) 7 to 10:

Step 1: From 7 to 10, the hour hand moves 3 hours.
Step 2: 3/12 = 1/4 of a revolution.

Answer1/4 (quarter revolution)

(d) 12 to 9:

Step 1: From 12 to 9, the hour hand moves 9 hours.
Step 2: 9/12 = 3/4 of a revolution.

Answer3/4 (three-quarter revolution)

(e) 1 to 10:

Step 1: From 1 to 10, the hour hand moves 9 hours.
Step 2: 9/12 = 3/4 of a revolution.

Answer3/4 (three-quarter revolution)

Question 2: Where will the hand of a clock stop if it: (a) Starts at 12 and makes 1/2 of a revolution, clockwise? (b) Starts at 2 and makes 1/2 of a revolution, clockwise? (c) Starts at 5 and makes 1/4 of a revolution, clockwise?

Solution

(a) Starts at 12, makes 1/2 revolution:

Step 1: Half revolution = 6 hours.
Step 2: 12 + 6 = 6 (which is 6 o'clock position).

AnswerThe hand stops at 6.

(b) Starts at 2, makes 1/2 revolution:

Step 1: Half revolution = 6 hours.
Step 2: 2 + 6 = 8.

AnswerThe hand stops at 8.

(c) Starts at 5, makes 1/4 revolution:

Step 1: Quarter revolution = 3 hours.
Step 2: 5 + 3 = 8.

AnswerThe hand stops at 8.

Question 3: Which direction will you face if you start facing: (a) East and make 1/2 of a revolution clockwise? (b) East and make 1-1/2 of a revolution clockwise? (c) West and make 3/4 of a revolution anti-clockwise?

Solution

Recall the directions: North, East, South, West go clockwise in order. Each turn is 1/4 revolution.

(a) East, 1/2 revolution clockwise:

Step 1: From East, half revolution = 2 quarter turns.
Step 2: East to South (1/4) to West (1/2).

AnswerYou face West.

(b) East, 1-1/2 revolutions clockwise:

Step 1: 1-1/2 = 3/2 revolutions = 1 full revolution + 1/2 revolution.
Step 2: 1 full revolution brings you back to East. Then 1/2 from East = West.

AnswerYou face West.

(c) West, 3/4 revolution anti-clockwise:

Step 1: Anti-clockwise from West: West to South (1/4), South to East (2/4), East to North (3/4).

AnswerYou face North.

Exercise 5.3 — Measuring Angles

Question 1: Measure the following angles using a protractor: (a) angle PQR (b) angle ABC (c) angle LMN

Solution

Step 1: Place the protractor so its centre is on the vertex of the angle.
Step 2: Align the base line of the protractor with one ray of the angle.
Step 3: Read the degree marking where the other ray crosses the protractor scale.
Answer (example values):
(a) Angle PQR = 45 degrees (acute angle)
(b) Angle ABC = 120 degrees (obtuse angle)
(c) Angle LMN = 90 degrees (right angle)

Question 2: Classify each angle as acute, obtuse, right, straight, or reflex: (a) 30 degrees (b) 90 degrees (c) 120 degrees (d) 180 degrees (e) 270 degrees (f) 75 degrees

Solution

Step 1: Use the definitions: Acute < 90, Right = 90, Obtuse > 90 and < 180, Straight = 180, Reflex > 180 and < 360.

Answer

(a) 30 degrees — Acute angle (less than 90 degrees)
(b) 90 degrees — Right angle (exactly 90 degrees)
(c) 120 degrees — Obtuse angle (between 90 and 180 degrees)
(d) 180 degrees — Straight angle (exactly 180 degrees)
(e) 270 degrees — Reflex angle (between 180 and 360 degrees)
(f) 75 degrees — Acute angle (less than 90 degrees)

Question 3: Which angle is greater: angle A = 45 degrees or angle B = 120 degrees?

Solution

Step 1: Compare the measures. 120 degrees > 45 degrees.

AnswerAngle B (120 degrees) is greater than Angle A (45 degrees).

Exercise 5.4 — Perpendicular Lines

Question 1: In the figure, are the lines perpendicular to each other?

Solution

Step 1: Two lines are perpendicular if they intersect at 90 degrees.
Step 2: Use a protractor to check the angle at the intersection.
Step 3: If the angle is exactly 90 degrees, the lines are perpendicular.

AnswerIf the angle at the intersection is 90 degrees, the lines are perpendicular. If not, they are not perpendicular.

Question 2: Which of the following are models of perpendicular lines: (a) Railway track (b) The edges of a blackboard (c) The letter L (d) The letter T?

Solution

Step 1: Perpendicular lines meet at exactly 90 degrees.
(a) Railway track — the two rails are parallel, not perpendicular.
(b) Edges of a blackboard — adjacent edges meet at 90 degrees. Yes, perpendicular.
(c) The letter L — the vertical and horizontal strokes meet at 90 degrees. Yes, perpendicular.
(d) The letter T — the top stroke meets the vertical stroke at 90 degrees. Yes, perpendicular.

Answer(b), (c), and (d) are models of perpendicular lines. (a) is not.

Question 3: The perpendicular bisector of a line segment is a line that divides the line segment into two equal parts at 90 degrees. In the figure, line l is the perpendicular bisector of segment AB. If AO = 3 cm, find AB.

Solution

Step 1: Since l is the perpendicular bisector of AB, O is the midpoint of AB.
Step 2: AO = OB = 3 cm (the bisector divides AB equally).
Step 3: AB = AO + OB = 3 + 3 = 6 cm.

AnswerAB = 6 cm.

Exercise 5.5 — Classification of Triangles

Question 1: Classify the following triangles based on their sides: (a) Triangle with sides 3 cm, 4 cm, 5 cm (b) Triangle with sides 5 cm, 5 cm, 8 cm (c) Triangle with sides 6 cm, 6 cm, 6 cm

Solution

Step 1: Recall the types: Scalene (all sides different), Isosceles (two sides equal), Equilateral (all three sides equal).

(a) Sides 3 cm, 4 cm, 5 cm:

All three sides are different.

AnswerScalene triangle.

(b) Sides 5 cm, 5 cm, 8 cm:

Two sides are equal (5 cm = 5 cm).

AnswerIsosceles triangle.

(c) Sides 6 cm, 6 cm, 6 cm:

All three sides are equal.

AnswerEquilateral triangle.

Question 2: Classify the following triangles based on their angles: (a) Triangle with angles 50 degrees, 60 degrees, 70 degrees (b) Triangle with angles 90 degrees, 30 degrees, 60 degrees (c) Triangle with angles 120 degrees, 30 degrees, 30 degrees

Solution

Step 1: Recall the types: Acute-angled (all angles < 90), Right-angled (one angle = 90), Obtuse-angled (one angle > 90).

(a) Angles 50, 60, 70 degrees:

All angles are less than 90 degrees.

AnswerAcute-angled triangle.

(b) Angles 90, 30, 60 degrees:

One angle is exactly 90 degrees.

AnswerRight-angled triangle.

(c) Angles 120, 30, 30 degrees:

One angle is greater than 90 degrees (120 degrees).

AnswerObtuse-angled triangle.

Question 3: Can a triangle have two right angles? Justify your answer.

Solution

Step 1: The sum of angles in a triangle is always 180 degrees.
Step 2: If two angles are 90 degrees each, their sum = 90 + 90 = 180 degrees.
Step 3: The third angle would be 180 - 180 = 0 degrees, which is impossible for a triangle.

AnswerNo, a triangle cannot have two right angles because the sum of all three angles must be exactly 180 degrees.

Exercise 5.6 — Quadrilaterals

Question 1: Name the quadrilateral whose: (a) All sides are equal and all angles are 90 degrees (b) Opposite sides are equal and all angles are 90 degrees (c) Opposite sides are parallel and equal but angles are not 90 degrees

Solution

(a) All sides equal, all angles 90 degrees:

Step 1: A quadrilateral with 4 equal sides and 4 right angles is a square.

AnswerSquare.

(b) Opposite sides equal, all angles 90 degrees:

Step 1: A quadrilateral with opposite sides equal and all right angles is a rectangle.

AnswerRectangle.

(c) Opposite sides parallel and equal, angles not 90 degrees:

Step 1: A quadrilateral where opposite sides are parallel and equal, but angles are not right angles, is a parallelogram.

AnswerParallelogram.

Question 2: How many diagonals does a quadrilateral have? Are they equal?

Solution

Step 1: A diagonal joins two non-adjacent vertices.
Step 2: In a quadrilateral ABCD, the diagonals are AC and BD.
Step 3: There are exactly 2 diagonals.
Step 4: Diagonals are equal in a rectangle and a square, but not in a general parallelogram or rhombus.

AnswerA quadrilateral has 2 diagonals. They are equal only in a rectangle and a square, but not necessarily in other quadrilaterals.

Question 3: State True or False: (a) A square is a rectangle. (b) A rectangle is a square. (c) A parallelogram is a quadrilateral.

Solution

(a) A square is a rectangle — TRUE.

Step 1: A rectangle has opposite sides equal and all angles 90 degrees.
Step 2: A square has all sides equal and all angles 90 degrees.
Step 3: Since a square satisfies all conditions of a rectangle, it is a special rectangle.

AnswerTrue.

(b) A rectangle is a square — FALSE.

Step 1: A square requires all four sides to be equal.
Step 2: A rectangle only requires opposite sides to be equal, not all sides.

AnswerFalse. A rectangle need not have all sides equal.

(c) A parallelogram is a quadrilateral — TRUE.

Step 1: A parallelogram has 4 sides and is a closed figure.
Step 2: Any four-sided closed figure is a quadrilateral.

AnswerTrue.

Exercise 5.7 — Three-Dimensional Shapes

Question 1: Match the following: (a) Cuboid (b) Sphere (c) Cylinder (d) Cone (e) Pyramid — with (i) A gas pipe (ii) A brick (iii) A ball (iv) A tent with triangular faces (v) A sweet laddu

Solution

Step 1: Identify the 3D shape of each object.

Answer

(a) Cuboid matches with (ii) A brick
(b) Sphere matches with (iii) A ball / (v) A sweet laddu (both are spherical)
(c) Cylinder matches with (i) A gas pipe
(d) Cone — not directly listed, but could match a conical object
(e) Pyramid matches with (iv) A tent with triangular faces

Question 2: Count the number of faces, edges, and vertices of: (a) A cube (b) A cuboid (c) A cylinder (d) A cone (e) A sphere

Solution

Shape	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Cylinder	3 (2 flat + 1 curved)	2	0
Cone	2 (1 flat + 1 curved)	1	1
Sphere	1 (curved)	0	0

Question 3: What is the shape of: (a) A brick (b) A road roller (c) A matchbox (d) A laddu?

Solution

Step 1: Identify the 3D shape of each everyday object.

Answer

(a) A brick — Cuboid
(b) A road roller — Cylinder
(c) A matchbox — Cuboid
(d) A laddu — Sphere

Exercise 5.8 — Polygons

Question 1: Identify which of the following are polygons: (a) A circle (b) A triangle (c) A square (d) A figure with curved sides

Solution

Step 1: A polygon is a closed figure made entirely of straight line segments (no curves).
(a) Circle — made of a curved line. Not a polygon.
(b) Triangle — made of 3 straight sides. Is a polygon.
(c) Square — made of 4 straight sides. Is a polygon.
(d) Figure with curved sides — has curves. Not a polygon.

Answer(b) Triangle and (c) Square are polygons. (a) Circle and (d) curved figure are not.

Question 2: Name each polygon by the number of sides: (a) 3 sides (b) 4 sides (c) 5 sides (d) 6 sides (e) 8 sides

Solution

Step 1: Match each number of sides to the polygon name.

Answer

(a) 3 sides — Triangle
(b) 4 sides — Quadrilateral
(c) 5 sides — Pentagon
(d) 6 sides — Hexagon
(e) 8 sides — Octagon

Question 3: How many diagonals does a hexagon have?

Solution

Step 1: The formula for diagonals of a polygon with n sides is: n(n - 3)/2.
Step 2: For a hexagon, n = 6.
Step 3: Number of diagonals = 6(6 - 3)/2 = 6 x 3 / 2 = 18/2 = 9.

AnswerA hexagon has 9 diagonals.

Common Mistakes

1. Confusing angle types. A right angle is exactly 90 degrees — not "about 90" or "close to 90." An acute angle is strictly less than 90, and an obtuse angle is strictly between 90 and 180.

2. Forgetting that a square is a special rectangle and a special parallelogram. A square satisfies all conditions of a rectangle (opposite sides equal, all angles 90) and a parallelogram (opposite sides parallel), but with the extra property that all sides are equal.

3. Measuring angles incorrectly with a protractor. Always place the protractor centre on the vertex and the base line on one ray. Read from 0 on the correct side (inner or outer scale depending on the ray's direction).

4. Saying a cylinder has no edges. A cylinder has 2 edges (the circular rims where the flat faces meet the curved surface). These are not sharp edges like a cube, but they are still edges.

5. Confusing the number of diagonals. A triangle has 0 diagonals, a quadrilateral has 2, and a pentagon has 5. Use the formula n(n - 3)/2 rather than guessing.

6. Mixing up clockwise and anti-clockwise directions in angle/revolution problems. Always check the direction specified. Clockwise goes in the same direction as clock hands; anti-clockwise is the opposite.

Exam Checklist

[ ] I can measure a line segment accurately using a ruler or divider.
[ ] I can calculate fractions of a revolution (e.g., 3 to 9 on a clock = 1/2 revolution).
[ ] I can classify angles as acute, right, obtuse, straight, reflex, or complete based on their degree measure.
[ ] I can measure an angle using a protractor correctly.
[ ] I can identify perpendicular lines and the perpendicular bisector of a line segment.
[ ] I can classify triangles by sides (scalene, isosceles, equilateral) and by angles (acute-angled, right-angled, obtuse-angled).
[ ] I can identify and describe quadrilaterals: rectangle, square, parallelogram, rhombus, trapezium.
[ ] I can count the faces, edges, and vertices of 3D shapes (cube, cuboid, cylinder, cone, sphere).
[ ] I can identify whether a figure is a polygon and name it by the number of sides.
[ ] I can use the formula n(n - 3)/2 to find the number of diagonals in a polygon.

Frequently Asked Questions

Use a ruler or divider to measure the distance between two endpoints. Align the ruler at zero with one endpoint.

NCERT Solutions for Class 6 Maths Chapter 5 Understanding Elementary Shapes | KalviGavani

NCERT Solutions for Class 6 Maths Chapter 5 — Understanding Elementary Shapes

Key Concepts

Exercise 5.1 — Measuring Line Segments

Exercise 5.2 — Angles: Right, Straight, and Complete

Exercise 5.3 — Measuring Angles

Exercise 5.4 — Perpendicular Lines

Exercise 5.5 — Classification of Triangles

Exercise 5.6 — Quadrilaterals

Exercise 5.7 — Three-Dimensional Shapes

Exercise 5.8 — Polygons

Common Mistakes

Exam Checklist

Frequently Asked Questions