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Shapes and Patterns

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 students to the world of shapes, patterns, and tessellations through hands-on activities like weaving paper mats and exploring geometric figures. It explains how patterns repeat in rows, how different shapes fit together without gaps (tessellation), and the properties of triangles, quadrilaterals, and other polygons. The one big idea a student must walk away with is that shapes have specific properties (equal sides, equal angles) that determine whether they can tile a surface or form patterns, and that these properties can be observed, described, and used to create designs.
Key Concepts (16)
Weaving pattern
A pattern created by weaving strips of paper over and under each other in a repeating sequence.
e.g. Row 1: 1 under, 1 over, 1 under, 1 over, ... (repeat). Row 2: 1 over, 1 under, 1 over, 1 under, ... (repeat).
Regular pentagon
A five-sided figure in which all sides are equal.
e.g. The pentagon shown in the chapter has all sides equal, so it is a regular pentagon.
Tessellation
Fitting shapes together around a point without any gaps or overlaps.
e.g. Regular triangles (equilateral triangles) fit together around a point without gaps or overlaps, so they tessellate.
Equilateral triangle
A triangle with all three sides equal.
e.g. The triangle given at the end of the book is an equilateral triangle.
Regular hexagon
A six-sided shape with all sides equal.
e.g. Regular hexagons fit together around a point without gaps or overlaps.
Regular octagon
A shape with eight equal sides.
e.g. Regular octagons do not tessellate because they leave gaps when fitted together.
Rhombus
A shape with all sides equal.
e.g. The rhombus in the chapter is divided into four triangles.
Isosceles triangle
A triangle with two equal sides.
e.g. When you cut the rhombus into triangles, each triangle has two equal sides, so they are isosceles triangles.
Scalene triangle
A triangle with no equal sides.
e.g. A triangle where all three sides are unequal is a scalene triangle.
Kite
A four-sided shape (quadrilateral) with two pairs of equal adjacent sides.
e.g. Side 1 = Side 2, and Side 3 = Side 4 in a kite.
Parallelogram
A quadrilateral whose opposite sides are equal.
e.g. Quadrilateral A and B in the chapter have opposite sides equal, so they are parallelograms.
Rectangle
A parallelogram in which all angles are equal and are right angles.
e.g. Quadrilateral B in the chapter has all angles equal and right angles, so it is a rectangle.
Tangram
A set of shapes cut from a square, used to form different figures.
e.g. The tangram set at the end of the textbook includes triangles, a square, and a parallelogram.
Icosahedron
A solid shape with 20 faces (triangles).
e.g. The icosahedron model made from the net has 20 triangular faces.
Dodecahedron
A solid shape with 12 faces (pentagons).
e.g. The dodecahedron model made from the net has 12 pentagonal faces.
Net of a solid
A flat pattern that can be folded to form a 3D shape.
e.g. The nets provided at the end of the book are used to make icosahedron and dodecahedron models.
Formulas & Equations (1)
No formulas or equations are present in this chapter.
Reads as: Not applicable
Variables: Not applicable
Use when: Not applicable
⚠ Not applicable
How It Works
Making a paper mat by weaving strips
  1. Step 1 — Take a coloured paper 30 cm long and 20 cm wide.
  2. Step 2 — Fold the coloured paper in half along the longer side.
  3. Step 3 — Draw vertical lines at equal distances from the closed end and cut slits leaving a gap of 3 cm at the top.
  4. Step 4 — Carefully unfold the paper. There will be no cuts in the paper at the top and the bottom.
  5. Step 5 — Cut 8 paper strips of 3 cm width in 2 colours and of length slightly longer than 20 cm.
  6. Step 6 — Take one colour strip and weave it across the slits going 1 under and 1 over, and again 1 under and 1 over. Repeat it for the first row.
  7. Step 7 — Take one more strip of another colour and weave it across the slits going 1 over and 1 under, and again 1 over and 1 under. Repeat it for the second row.
  8. Step 8 — Weave all the strips in the same alternating pattern. Neatly fold any extra strip ends behind the mat.
💡 The pattern alternates: Row 1 is 1 under, 1 over; Row 2 is 1 over, 1 under. Students often forget to alternate the starting pattern for each row.
Checking if a shape tessellates
  1. Step 1 — Take cutouts of the shape (e.g., regular pentagon, equilateral triangle, square, regular hexagon, regular octagon).
  2. Step 2 — Try to fit the shapes together around a point without any gaps or overlaps.
  3. Step 3 — Count how many shapes fit around the point.
  4. Step 4 — If they fit without gaps or overlaps, the shape tessellates. If gaps remain, it does not tessellate.
💡 Regular pentagons and regular octagons do not tessellate; equilateral triangles, squares, and regular hexagons do tessellate.
Making a kite from a square piece of paper
  1. Step 1 — Start with a square piece of paper.
  2. Step 2 — Take one corner of the paper and fold it towards the opposite corner, creating a sharp crease along the diagonal.
  3. Step 3 — Open and fold the corner A inwards, aligning the edge with the crease you just made.
  4. Step 4 — Repeat on the other side, folding the other corner B inwards to align with the crease at the centre.
💡 The final shape is a kite. Students often forget to align the edge exactly with the central crease.
Creating circle designs by joining points
  1. Step 1 — Draw a circle and mark points 1 to 24 around it.
  2. Step 2 — Join points 1 to 11, 11 to 2, 2 to 12, and so on until you reach back at 1.
  3. Step 3 — Use different coloured threads on a thick paper or cloth to create the design.
💡 The pattern repeats in a specific sequence. Students often join points in the wrong order.
Finding how many small cubes have painted faces
  1. Step 1 — A large cube is made by gluing 27 small cubes together (3x3x3).
  2. Step 2 — The large cube is painted red on all outer faces.
  3. Step 3 — Count cubes with three faces painted (corner cubes): 8.
  4. Step 4 — Count cubes with two faces painted (edge cubes, excluding corners): 12.
  5. Step 5 — Count cubes with one face painted (centre of each face): 6.
  6. Step 6 — Count cubes with no faces painted (inner cube): 1.
💡 Students often forget that the centre cube (1) has no painted faces, and they miscount the edge cubes.
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'regular pentagon' with 'pentagon'
✗ A pentagon has all sides equal.
→ Not all pentagons have equal sides; only regular pentagons have all sides equal.
✓ A regular pentagon is a five-sided figure in which all sides are equal.
⚠ Thinking all triangles tessellate
✗ All triangles tessellate.
→ Only equilateral triangles (regular triangles) tessellate without gaps or overlaps. Scalene or isosceles triangles may not fit perfectly around a point.
✓ Equilateral triangles tessellate because they fit together around a point without gaps or overlaps.
⚠ Mixing up kite and parallelogram properties
✗ A kite has opposite sides equal.
→ A kite has two pairs of equal adjacent sides, not opposite sides. Opposite sides are equal in a parallelogram.
✓ A kite has two pairs of equal adjacent sides (Side 1 = Side 2, Side 3 = Side 4).
⚠ Calling a rectangle just a 'rectangle' without linking to parallelogram
✗ A rectangle is a four-sided shape with right angles.
→ This definition is incomplete. A rectangle is a special type of parallelogram where all angles are equal and are right angles.
✓ A rectangle is a parallelogram in which all angles are equal and are right angles.
⚠ Incorrectly counting painted faces on a cube
✗ A 3x3x3 cube has 8 cubes with three faces painted, 8 with two faces painted, and 6 with one face painted.
→ The number of cubes with two faces painted is 12 (each edge has 1 centre cube, and there are 12 edges). Students often count 8 by mistake.
✓ For a 3x3x3 cube: three faces painted = 8 (corners), two faces painted = 12 (edge centres), one face painted = 6 (face centres), no faces painted = 1 (inner cube).
Compare & Contrast
Number of equal sides
Equilateral triangle
Isosceles triangle
An equilateral triangle has all three sides equal, while an isosceles triangle has exactly two equal sides.
Angles
Parallelogram
Rectangle
A parallelogram has opposite angles equal, but they are not necessarily right angles. A rectangle is a special parallelogram where all angles are equal and are right angles.
Tessellation ability
Regular pentagon
Regular hexagon
Regular pentagons do not tessellate (they leave gaps), while regular hexagons do tessellate (they fit together without gaps or overlaps).
Exam Focus
Marks pattern: 2 MCQs (1 mark each) on shape properties, 1 SA (2 marks) on tessellation, 1 LA (4 marks) on cube painting or pattern description
Most asked: Properties of triangles (equilateral, isosceles, scalene) · Properties of quadrilaterals (kite, parallelogram, rectangle) · Tessellation and which shapes tessellate · Counting painted faces on a cube
Don't skip: Describe the properties of a kite and a parallelogram.; Explain why regular pentagons do not tessellate but equilateral triangles do.; A 3x3x3 cube is painted. How many small cubes have 3, 2, 1, or 0 faces painted?
One-Line Revision (read aloud before the exam)
  1. A regular pentagon has all five sides equal.
  2. Tessellation means fitting shapes together around a point without gaps or overlaps.
  3. Equilateral triangles, squares, and regular hexagons tessellate.
  4. Regular pentagons and regular octagons do not tessellate.
  5. An equilateral triangle has all three sides equal.
  6. An isosceles triangle has two equal sides.
  7. A scalene triangle has no equal sides.
  8. A kite has two pairs of equal adjacent sides.
  9. A parallelogram has opposite sides equal.
  10. A rectangle is a parallelogram with all right angles.
  11. In a 3x3x3 painted cube, 8 small cubes have three faces painted, 12 have two faces painted, 6 have one face painted, and 1 has no faces painted.
  12. A rhombus has all sides equal.
  13. A tangram is a set of shapes cut from a square used to form different figures.
  14. An icosahedron has 20 triangular faces.
  15. A dodecahedron has 12 pentagonal faces.

MCQs — 10 Questions with Answers

Q1. What does tessellation mean?

  • A. Shapes fitting together with no gaps or overlaps ✓
  • B. Shapes arranged in a line
  • C. Shapes of different colours
  • D. Shapes that are very small

Answer: A — Tessellation is when shapes fit perfectly together to cover a surface completely without any gaps between them or overlapping on top of each other.

Q2. Which of these shapes can tessellate?

  • A. Regular pentagon
  • B. Regular octagon
  • C. Equilateral triangle ✓
  • D. Irregular hexagon

Answer: C — Equilateral triangles tessellate because six of them fit perfectly around a point with no gaps, but pentagons and octagons leave gaps.

Q3. An isosceles triangle has _____ equal sides.

  • A. One
  • B. Two ✓
  • C. Three
  • D. No

Answer: B — An isosceles triangle has exactly two equal sides and also two equal angles, making it different from equilateral which has three equal sides.

Q4. How many equilateral triangles fit around a point without gaps?

  • A. Four
  • B. Five
  • C. Six ✓
  • D. Eight

Answer: C — Six equilateral triangles fit perfectly around a central point with no gaps or overlaps because each angle of an equilateral triangle is 60°, and 6 × 60° = 360°.

Q5. Which shape is a kite?

  • A. A quadrilateral with all sides equal
  • B. A quadrilateral with 2 pairs of adjacent equal sides ✓
  • C. A quadrilateral with opposite sides equal
  • D. A quadrilateral with no equal sides

Answer: B — A kite has two pairs of adjacent equal sides, meaning sides next to each other are equal (sides 1=2 and sides 3=4), not opposite sides.

Q6. A rectangle is a special type of _____ with all right angles.

  • A. Triangle
  • B. Pentagon
  • C. Parallelogram ✓
  • D. Kite

Answer: C — A rectangle is a special parallelogram because it has opposite sides equal like a parallelogram, but all its angles are 90° right angles.

Q7. In a weaving pattern, if Row 1 is '1 under, 1 over', what should Row 2 be to create an alternating pattern?

  • A. 1 under, 1 over
  • B. 1 over, 1 under ✓
  • C. 2 under, 2 over
  • D. 1 under only

Answer: B — Row 2 must reverse the pattern of Row 1 by doing '1 over, 1 under' so that the strips weave over and under alternately to create the interlocking pattern.

Q8. A scalene triangle has _____ equal sides.

  • A. All three
  • B. Two
  • C. One
  • D. No ✓

Answer: D — A scalene triangle has no equal sides and no equal angles, making it different from isosceles (2 equal sides) and equilateral (3 equal sides).

Q9. Which of these can fit together around a point with no gaps? [Looking at: square, regular pentagon, regular hexagon, regular octagon]

  • A. Square and regular pentagon
  • B. Square and regular octagon ✓
  • C. Regular pentagon and octagon
  • D. All four shapes

Answer: B — Four squares fit around a point (4 × 90° = 360°) and three regular hexagons fit around a point (3 × 120° = 360°), but pentagons and octagons do not tessellate alone or together.

Q10. In a parallelogram, which sides are equal?

  • A. All four sides
  • B. Adjacent sides
  • C. Opposite sides ✓
  • D. Only two sides

Answer: C — A parallelogram has opposite sides equal, meaning the top and bottom sides are equal to each other, and the left and right sides are equal to each other.

Flashcards

What is tessellation?

Tessellation means shapes fit together perfectly to cover a surface with no gaps or overlaps.

Which shape tessellates: regular pentagon or regular triangle?

Regular triangles tessellate because they fit around a point with no gaps, but regular pentagons do not.

What is an equilateral triangle?

An equilateral triangle is a triangle with all three sides equal and all three angles equal.

What is an isosceles triangle?

An isosceles triangle is a triangle with exactly two equal sides and two equal angles.

What is a scalene triangle?

A scalene triangle is a triangle with no equal sides and no equal angles.

What is a kite in geometry?

A kite is a quadrilateral with two pairs of adjacent equal sides.

What is a parallelogram?

A parallelogram is a quadrilateral where opposite sides are equal and opposite angles are equal.

What is a rectangle?

A rectangle is a special parallelogram with all angles as right angles and opposite sides equal.

In weaving pattern Row 1: 1 under, 1 over, what comes in Row 2?

In Row 2, the pattern reverses: 1 over, 1 under, repeating to create an alternating weave.

How many regular triangles fit around a point?

Six equilateral triangles fit perfectly around a point with no gaps or overlaps.

Important Board Questions

What is tessellation? Give one example from daily life. [1 mark]

Tessellation means shapes fit with no gaps or overlaps. Think of floor tiles, ceiling patterns, or basket weaving at home.

Draw and label an equilateral triangle and an isosceles triangle. How are they different? [2 marks]

Equilateral has 3 equal sides and 3 equal angles. Isosceles has only 2 equal sides and 2 equal angles. Show this with your drawings and mark the equal sides.

Explain with an example how a weaving pattern works. Write the pattern for two rows. [3 marks]

Describe how strips go under and over in alternating rows. Row 1 goes 1 under, 1 over; Row 2 reverses to 1 over, 1 under. Show this pattern repeats to create the final woven design.

A kite and a parallelogram are both quadrilaterals with some equal sides. Explain how their equal sides are different. Draw both shapes and mark the equal sides. [5 marks]

In a kite, equal sides are adjacent (next to each other): sides 1=2 and sides 3=4. In a parallelogram, equal sides are opposite (across from each other): top=bottom and left=right. Draw both clearly and label or colour the equal pairs differently.

True or False: A rectangle is a special type of parallelogram. Give a reason for your answer. [2 marks]

A rectangle has all the properties of a parallelogram (opposite sides equal, opposite angles equal) PLUS all angles are 90° right angles, making it special.

Assertion-Reason Questions for Shapes and Patterns

Q1. Assertion (A):

Equilateral triangles can be used to tessellate a floor without any gaps.

Reason (R):

Equilateral triangles have all sides equal and when 6 of them fit around a point, they leave no gaps.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why equilateral triangles tessellate — 6 triangles fit perfectly around a point.

Q2. Assertion (A):

A rectangle is a special type of parallelogram.

Reason (R):

Both rectangles and parallelograms have opposite sides equal.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true and the reason is also true, but the reason is not the complete explanation. A rectangle is special because it also has all right angles, which not all parallelograms have.

Q3. Assertion (A):

Regular pentagons can tessellate perfectly to cover a floor.

Reason (R):

When three pentagons are arranged around a point, there is always an empty space that cannot fit another pentagon.

Answer: D — A is false but R is true — The assertion is false. Regular pentagons do NOT tessellate because they leave gaps. The reason correctly explains why they don't tessellate, but the assertion itself is wrong.

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Case-Based Questions (CBQ) for Shapes and Patterns

Case Study:

Meera's grandmother is making a woven mat for their living room. She takes a colored paper sheet 30 cm long and 20 cm wide. She folds it in half along the longer side and cuts vertical slits, leaving a 3 cm gap at the top. After unfolding, she weaves colored paper strips through the slits, alternating the pattern — first row goes under-over-under-over, and the second row goes over-under-over-under.

Q1. What should Meera's grandmother do before cutting the slits?

  • A. Fold the paper in half ✓
  • B. Cut the paper into strips
  • C. Color the paper
  • D. Measure only the width

Answer: A

Q2. Why is the 3 cm gap at the top important?

  • A. To make the mat look pretty
  • B. To keep the paper from falling apart when unfolded ✓
  • C. To match the width of the strips
  • D. To make folding easier

Answer: B

Q3. In the second row of weaving, the pattern is 'over-under-over-under.' This is different from the first row because it creates which type of pattern?

  • A. A straight line pattern
  • B. An alternating weaving pattern ✓
  • C. A random pattern
  • D. A spiral pattern

Answer: B

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