Class 5 Mathematics

Chapter 7 Solutions — Shapes and Patterns

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Solutions

Overview

Step-by-step NCERT solutions for Shapes and Patterns (Chapter 7, NCERT Class 5 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Shapes and Patterns textbook chapter.

Solved

What these solutions cover

All 32 questions in Shapes and Patterns are solved in the PDF. Here's what's inside, exercise by exercise:

Weaving Mats — Paper Mat Activities (p. 92–93)

1.
Let us make paper mats. Follow the steps:
- (a) Take a coloured paper 30 cm long and 20 cm wide.
- (b) Fold it in half along the longer side.
- (c) Draw vertical lines at equal distances from the closed end and cut slits, leaving a gap of 3 cm at the top.
- (d) Carefully unfold the paper.
- (e) Cut 8 paper strips of 3 cm width in 2 colours, slightly longer than 20 cm.
- (f) Weave the first strip going 1…
Can you figure out how to make this mat? The weaving pattern for the first two rows is: Row 1 — 2 over, 1 under, 2 over, 1 under, ... Row 2 — 2 under, 1 over, 2 under, 1 over, ...
Try to weave a pattern using the rules given below. Row 1 — 2 over, 1 under, 2 over, 1 under, ... (repeat) Row 2 — 1 under (do not repeat), 3 over, 3 under, 3 over, 3 under, ... (repeat) Row 3 — 2 under, 1 over, 2 under, ... (repeat) Row 4 — 1 over (do not repeat), 3 under, 3 over, ... (repeat) Continue weaving in this order.
Can you work out the steps for any of these designs and weave the pattern? Write the steps of the pattern in your notebook for each row until it starts repeating. (Two woven mat designs with diagonal brick-like patterns are shown.)

Let Us Try — Grid Pattern Drawing (p. 93)

Draw the following pattern on a grid paper. Part of it is done for you. Complete the rest of the grid to get the full design. (A woven mat pattern is shown partially filled in a dotted grid.)

Tiling and Tessellation / Find Out — Regular Shapes (p. 94–95)

Can regular triangles (equilateral triangles) fit together at a point without any gap? How many of them fit together? (Use the sample triangle from the end of the book.)
Can squares (regular 4-sided shapes) fit together around a point without any gap or overlap? How many squares fit together at a point? Can five squares fit together at a point without gaps or overlaps? Why or why not?
Can regular hexagons (6-sided shapes with equal sides) fit together around a point without any gaps or overlaps? How many fit together at a point?
Here is a tessellating pattern with more than one shape. What shapes have been used in this pattern? (A pattern using two shapes — large hexagon-like shapes and triangles — is shown.)
Continue the pattern given below and colour it appropriately. (A partly drawn pattern of hexagons and triangles is shown on a dotted grid.)
Look at the pattern given below. What shapes are coming together at the marked points? Are the same set of shapes coming together at these points? Continue the pattern and colour it appropriately. (A pattern of regular octagons and small squares on a dotted grid is shown, with two points marked.)
Here is a tiling pattern made using two different shapes — squares and triangles. Are the triangles equilateral? Why or why not? What shapes are coming together at the marked points? Are the same set of shapes coming together at these points? Continue the pattern and colour it appropriately.

Rhombus Triangle Activities — Triangles and Quadrilaterals (p. 96–98)

What geometrical shapes can you make by fitting 2 triangles (from the rhombus cutout) together? Trace the shapes you created. How many different types of triangles can you make? Observe and measure the sides of these triangles. What do you notice?
Is it possible to make a triangle where all three sides are equal (equilateral triangle) using the rhombus triangles?
Is it possible to make a triangle where all three sides are unequal (scalene triangle) using the rhombus triangles?
How many different 4-sided shapes (quadrilaterals) can you make using 2 triangles from the rhombus? What do you notice about the shape called a kite? (Side 1 = Side 2, Side 3 = Side 4 — these are adjacent sides.)
Measure the sides of each of the two quadrilaterals A and B (shown in the book). What do you notice? Are there any pairs of sides that are equal? Which pairs are equal — adjacent or opposite? What types of angles do A and B have? Which angles are equal in each?
In the grid given below, draw two different kites and two different parallelograms.
7.
Now use 3 triangles from the rhombus to form shapes. How many sides do each one have? Using 3 triangular pieces, try creating:
- (a) a 3-sided shape,
- (b) a 4-sided shape, and
- (c) a 5-sided shape.
8.
Which of these shapes can be made with all 4 triangular pieces from the rhombus? Try and find out.
- (a) Square
- (b) Rectangle
- (c) Triangle
- (d) Pentagon (5-sided)
- (e) Hexagon (6-sided)
- (f) Octagon (8-sided)

Try This — Equilateral and Scalene Triangles (p. 97)

Cut the equilateral triangle provided at the end of the book. Check if all the angles of an equilateral triangle are equal (by folding, as you did with the isosceles triangle). Then cut the equilateral triangle in half. How many sides of each new triangle are equal? What are scalene triangles?

Tangram — Shapes and Angles (p. 99)

1.
Look at the tangram set given at the end of your textbook. Cut out all the shapes and name them.
- (a) How are they same or different from each other?
- (b) What do you notice about the angles of each of the shapes?
- (c) What do you notice about the sides of each of the shapes?
Use some or all of the pieces of your tangram set to make the following shapes (shown in the book): a large triangle, a small triangle, a rectangle, and a trapezium. There may be more than one way to do it.

Which Shape Am I? — Matching Statements to Shapes (p. 100)

Match each statement with the correct shape(s) from: Square, Rectangle, Rhombus, Parallelogram. Some statements may describe more than one shape. 1. All my angles are right angles, but all my sides are not equal. 2. All my sides are equal, but all my angles are not. 3. My opposite angles are equal, but my sides do not make a right angle. 4. Two pairs of sides are equal, but they do not make a…

Kites — Paper Folding Activity (p. 100)

1.
Make your own kite shape from a square piece of paper by following the steps:
- (a) Start with a square piece of paper.
- (b) Fold it towards the opposite corner along the diagonal (crease along diagonal).
- (c) Open and fold corner A inwards, aligning the edge with the crease.
- (d) Repeat on the other side, folding corner B inwards to align with the crease. You have a kite shape! What shapes do you…

Play with Circles — Diameter and Quadrilateral (p. 101)

Draw a circle with a compass and mark its centre. Draw a diameter and mark both endpoints. Draw another diameter and mark its endpoints. Now join the four endpoints. What shape is formed? Check the sides and angles of the quadrilateral. Try with a different pair of diameters. What do you notice about the shape that is formed? Is it possible to create a 4-sided shape other than a rectangle…

Circle Designs — Thread Design on 24-Point Circle (p. 101)

Look at the circle marked with points 1 to 24. Join points 1 to 11, 11 to 2, 2 to 12, and so on till you reach back at 1. (Try it with different coloured threads on a thick paper or cloth.) Can you think of a way to make a design exactly like the image given (a flower pattern made of overlapping circles)?

Cube Connections

Here are three views of a cube (top view, front view, and side view — each face has a different design). Can you draw them on the net in the correct order?
2.
Here are some big solid cube frames. How many small cubes have been removed from each cube? (Three cubes are shown — each has a square tunnel drilled straight through from each of its three pairs of faces:
- (a) a 3x3x3 cube with a 1x1 tunnel,
- (b) a 4x4x4 cube with a 2x2 tunnel,
- (c) a 5x5x5 cube with a 3x3 tunnel.)
3.
Nisha has glued 27 small cubes together to make a large solid cube. She paints the large cube red. How many of the original small cubes have —
- (a) three faces painted red?
- (b) two faces painted red?
- (c) one face painted red?
- (d) no faces painted red?

Puzzle — Tanu's Shapes in a Line (p. 102)

1.
Tanu arranged 7 shapes in a line: 2 squares, 2 triangles, 1 circle, 1 hexagon, and 1 rectangle. Find her arrangement using these clues:
- (a) The square is between the circle and the rectangle.
- (b) The rectangle is between the square and the triangle.
- (c) The two triangles are next to the square.
- (d) The hexagon is to the right of the triangle.
- (e) The circle is to the left of the square.

Icosahedron and Dodecahedron — 3-D Shapes (p. 103)

Use the nets provided at the end of the book to make icosahedron and dodecahedron models. Then answer: — What shapes do you see in an icosahedron and a dodecahedron? — Do all the faces look the same? — How many faces meet at a vertex? — Do the same number of faces meet at each vertex? — How many edges do you see?

Sample

Worked examples

3 questions from Shapes and Patterns, solved step by step.

Tiling and Tessellation / Find Out — Regular Shapes (p. 94–95)

Can regular triangles (equilateral triangles) fit together at a point without any gap? How many of them fit together? (Use the sample triangle from the end of the book.)

Yes, regular (equilateral) triangles fit perfectly around a point without any gap or overlap.

6 equilateral triangles fit around a single point.

Why? Each angle of an equilateral triangle is 60 degrees.

6 x 60 = 360 degrees, which is exactly one full turn around a point.

So 6 triangles fit perfectly.

Answer: Yes. 6 equilateral triangles fit around a point with no gap. Each angle is 60 degrees, and 6 x 60 = 360 degrees (a full circle).

Rhombus Triangle Activities — Triangles and Quadrilaterals (p. 96–98)

What geometrical shapes can you make by fitting 2 triangles (from the rhombus cutout) together? Trace the shapes you created. How many different types of triangles can you make? Observe and measure the sides of these triangles. What do you notice?

Each piece cut from the rhombus is a triangle with two equal sides (an isosceles triangle).

Fit the two identical pieces together along different matching edges and trace what you get:

- Joined along the equal (slant) sides, the outer edges line up to form a larger triangle.

- Joined along the cut (diagonal) edge, you get back the rhombus (not a triangle).

By trying the different edges you can make more than one triangle. Measure the sides each time: the triangles you form are isosceles (two equal sides), so they also have two equal angles.

Answer: Fitting the two equal pieces along different edges makes more than one triangle (for example a larger isosceles triangle); joining along the cut edge gives back the rhombus. The triangles formed are isosceles - two equal sides and two equal angles. (Trace and measure to compare.)

Try This — Equilateral and Scalene Triangles (p. 97)

Cut the equilateral triangle provided at the end of the book. Check if all the angles of an equilateral triangle are equal (by folding, as you did with the isosceles triangle). Then cut the equilateral triangle in half. How many sides of each new triangle are equal? What are scalene triangles?

Checking angles of an equilateral triangle:

Fold the equilateral triangle so that two vertices meet. The fold line should pass through the third vertex. If the two halves match perfectly, the two base angles are equal. Repeat for another pair — all three angles are equal.

Result: Equilateral triangles have 3 equal angles.

Cutting the equilateral triangle in half:

When you cut an equilateral triangle in half (along the height from one vertex to the midpoint of the opposite side), you get two smaller triangles.

Each new triangle has: one short side (half the base), one medium side (the height), and one long side (a full side of the original equilateral triangle).

All three sides are different lengths — no two sides are equal.

Triangles with no equal sides are called scalene triangles.

In a scalene triangle, no two angles are equal either.

Answer: All three angles of an equilateral triangle are equal. When cut in half, each new triangle has 3 sides of different lengths — it is a scalene triangle. Scalene triangles have no equal sides and no equal angles.

All 32 questions in Shapes and Patterns are solved step by step in the Solutions PDF above.

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Chapter 7 Solutions — Shapes and Patterns

Overview

What these solutions cover

Weaving Mats — Paper Mat Activities (p. 92–93)

Let Us Try — Grid Pattern Drawing (p. 93)

Tiling and Tessellation / Find Out — Regular Shapes (p. 94–95)

Rhombus Triangle Activities — Triangles and Quadrilaterals (p. 96–98)

Try This — Equilateral and Scalene Triangles (p. 97)

Tangram — Shapes and Angles (p. 99)

Which Shape Am I? — Matching Statements to Shapes (p. 100)

Kites — Paper Folding Activity (p. 100)

Play with Circles — Diameter and Quadrilateral (p. 101)

Circle Designs — Thread Design on 24-Point Circle (p. 101)

Cube Connections

Puzzle — Tanu's Shapes in a Line (p. 102)

Icosahedron and Dodecahedron — 3-D Shapes (p. 103)

Worked examples

Can regular triangles (equilateral triangles) fit together at a point without any gap? How many of them fit together? (Use the sample triangle from the end of the book.)

What geometrical shapes can you make by fitting 2 triangles (from the rhombus cutout) together? Trace the shapes you created. How many different types of triangles can you make? Observe and measure the sides of these triangles. What do you notice?

Cut the equilateral triangle provided at the end of the book. Check if all the angles of an equilateral triangle are equal (by folding, as you did with the isosceles triangle). Then cut the equilateral triangle in half. How many sides of each new triangle are equal? What are scalene triangles?

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