Fun with Shapes — NCERT Solutions

Given: A dot grid is provided to recreate Amma's rangoli.

Activity: Using the dot grid as a guide, carefully join the dots to reproduce the same rangoli pattern as Amma's. Follow the same shapes (circles, triangles, squares, etc.) and their positions as shown in the original rangoli.

Note: This is a hands-on drawing activity. Trace or draw the rangoli pattern on the dot grid provided, matching the shapes and their arrangement.

Given: Amma's rangoli contains various geometric shapes.

Concept: A rangoli typically uses basic geometric shapes such as circles, triangles, squares, and rectangles.

Answer: The shapes drawn in Amma's rangoli are:
$$\text{Circle, Triangle, Rectangle (or Square)}$$

(Note: The exact shapes depend on the rangoli figure. Common shapes found in rangolis are circle, triangle, and square/rectangle.)

Given: Amma's rangoli contains shapes made with curved lines and straight lines.

Concept:

• Shapes made with curved lines: Circle, oval, etc.
• Shapes made with straight lines: Triangle, square, rectangle, etc.

Answer (based on a typical rangoli with 1 circle, 1 triangle, and 1 rectangle/square):
(a) Curved lines: $1$ (circle)
(b) Straight lines: $2$ (triangle and rectangle/square)

(Note: The exact count depends on the rangoli figure shown in the book.)

Given: A cardboard box (cuboid shape) is opened and flattened.

Concept: When a 3D box (cuboid) is opened and laid flat, it shows its net — the 2D faces of the box.

Observation: When a cardboard box is flattened, we see rectangles (and sometimes squares). A cuboid has 6 faces, all of which are rectangles (opposite faces are equal in size).

Answer: When a cardboard box is opened and flattened, we see rectangles (and squares) — the flat faces of the box.

Given: A square piece of paper and step-by-step folding instructions (shown in figures 1–4).

Steps to make an envelope:

1. Take a square piece of paper and place it like a diamond (rotated 45°).
2. Fold the bottom corner up to the centre of the square.
3. Fold the left and right corners inward to meet at the centre.
4. Fold the top corner down to close the envelope.

Result: You get a neat envelope shape. This is a hands-on paper-folding (origami) activity.

Given: A cuboidal object (like an eraser or sharpener) is used to trace its faces.

Concept: A cuboid has 6 faces. Opposite faces are equal rectangles. If all sides are equal, the faces are squares.

Answers:
(a) How many different faces did you get?
$$\text{A cuboid has 3 pairs of faces, so we get up to 3 different sizes of faces (6 faces total).}$$

(b) What shapes are these faces?
$$\text{The faces are rectangles (or squares).}$$

(c) Did you get a square?
$$\text{It depends on the object. If the cuboid has equal length and width on one face, that face is a square. For a regular eraser, some faces may be squares.}$$

(d) Can you get six different rectangles by tracing a cuboid?
$$\text{No. A cuboid has 3 pairs of identical faces, so opposite faces are the same. You get at most 3 different rectangles (each repeated twice).}$$

(e) Can a cuboid have a face like a triangle?
$$\text{No. A cuboid cannot have a triangular face. All its faces are rectangles or squares.}$$

(f) The faces of a cuboid are $\underline{\text{rectangle}}$ or $\underline{\text{square}}$ in shape.

Given: Sides (line segments) are provided to construct rectangles.

Concept: A rectangle has 4 sides — two pairs of equal opposite sides, and all corners are square corners (right angles).

Steps:

1. Take the given side as the length of the rectangle.
2. Draw a line equal to the given length.
3. At each end, draw a perpendicular line equal to the given width.
4. Join the ends to complete the rectangle.

Result: A rectangle is formed with the given sides. (This is a drawing/construction activity on the dot grid provided.)

Given: A small rectangle is drawn on a dot grid.

Concept: A bigger rectangle is one that has greater length and/or width than the given rectangle, with all square corners.

Steps:

1. Look at the given rectangle on the dot grid.
2. Draw a rectangle slightly bigger around it (keeping the same centre or corner).
3. Draw a second rectangle even bigger around the first.
4. Draw a third rectangle even bigger around the second.

Result: Three concentric (nested) rectangles of increasing size are drawn around the original rectangle. (This is a drawing activity on the dot grid.)

Given: A picture containing multiple rectangles (including overlapping ones) is shown.

Concept: Count all rectangles — small individual ones as well as larger ones formed by combining smaller rectangles.

Method: Count systematically — first count the smallest rectangles, then count rectangles made by combining 2, 3, or more smaller rectangles.

Answer: The exact number depends on the figure. (Note: The figure is not visible in the OCR. Students should count all individual and combined rectangles carefully in the picture provided in the book.)

Tip: A typical such figure with a $2 \times 3$ grid of rectangles would have $2 \times 3 + 1 \times 3 + 2 \times 2 + 1 \times 2 + 2 \times 1 + 1 \times 1 =$ several rectangles. Count carefully!

Given: Different rectangles of various sizes are shown.

Concept: Properties of a rectangle.

Answers:
(a) How many sides are there in a rectangle?
$$\text{A rectangle has } \mathbf{4} \text{ sides.}$$

(b) How many corners are there in a rectangle?
$$\text{A rectangle has } \mathbf{4} \text{ corners.}$$

(c) Are there any sides in a rectangle that are equal in length to each other?
$$\text{Yes. In a rectangle, the two opposite sides are equal in length.}$$
$$\text{The two longer sides (lengths) are equal, and the two shorter sides (widths) are equal.}$$

(d) What do you notice in a rectangle? Describe it in your own words.
$$\text{A rectangle has 4 sides and 4 corners. All 4 corners are square corners (right angles).}$$
$$\text{The opposite sides are equal in length. The two longer sides are equal and the two shorter sides are equal.}$$

Given: A square and a rectangle are being compared.

Concept: Both square and rectangle are quadrilaterals.

Answer:
$$\text{Both have } \mathbf{4} \text{ sides.}$$
$$\text{Both have } \mathbf{4} \text{ corners.}$$

Given: A drawing containing multiple squares of different sizes.

Concept: Count all squares — small individual ones and larger ones formed by combining smaller squares.

Method: Count systematically — first the smallest squares, then $2\times2$ squares, then $3\times3$ squares, etc.

Answer: The exact number depends on the figure in the book. (Students should count all individual and combined squares carefully.)

Example: In a $3 \times 3$ grid of unit squares:

• $1 \times 1$ squares: $9$
• $2 \times 2$ squares: $4$
• $3 \times 3$ squares: $1$
• Total: $9 + 4 + 1 = 14$ squares

Given: A small square is drawn on a dot grid.

Concept: A bigger square has all 4 equal sides longer than the original, with all square corners.

Steps:

1. Look at the given square on the dot grid.
2. Draw a square with sides 2 units longer (1 unit extra on each side) around the original.
3. Draw another square with sides even longer around the second square.

Result: Two larger squares are drawn concentrically around the original square. (This is a drawing activity on the dot grid.)

Given: Matchsticks are used to make a central square with squares on all its sides.

Concept: If we make one square in the centre and attach one square on each of its 4 sides, we use matchsticks to form a cross/plus shape.

Working:

• Central square: 4 matchsticks
• Square on top: 3 new matchsticks (shares 1 side with centre)
• Square on bottom: 3 new matchsticks
• Square on left: 3 new matchsticks
• Square on right: 3 new matchsticks
• Total matchsticks: $4 + 3 + 3 + 3 + 3 = 16$ matchsticks

Number of squares: $1$ (centre) $+ 4$ (on each side) $= \mathbf{5}$ squares total.

Given: One side of a square is provided; complete the square.

Concept: A square has all 4 sides equal and all 4 corners are square corners (right angles).

Steps:

1. Measure the length of the given side.
2. At each end of the given side, draw a perpendicular line of the same length.
3. Join the two free ends to complete the square.

Result: A complete square is formed. (This is a drawing/construction activity.)

Given: Square cutouts are used to form different shapes by joining them edge-to-edge.

Concept: Shapes made by joining squares edge-to-edge are called polyominoes.

Answers:
(a) Joining 2 squares (dominoes):
There is $\mathbf{1}$ different shape — two squares joined side by side (a $1 \times 2$ rectangle).

(b) Joining 3 squares (trominoes):
There are $\mathbf{2}$ different shapes:

• A straight row of 3 squares ($1 \times 3$ rectangle)
• An L-shape (2 squares in a row with 1 square on top of one end)

(c) Joining 4 squares (tetrominoes):
There are $\mathbf{5}$ different shapes:

• Straight line ($1 \times 4$)
• L-shape
• T-shape
• S/Z-shape
• Square ($2 \times 2$)

(Draw each of these on the dot grid provided.)

Given: Several shapes are shown; identify which are rectangles.

Concept: A rectangle has:

• 4 sides
• 4 square corners (right angles)
• Opposite sides equal in length

Answer: Tick all shapes that have 4 sides, 4 square corners, and opposite sides equal.

Shapes that are NOT rectangles:

• A triangle is not a rectangle because it has only 3 sides and 3 corners.
• A circle is not a rectangle because it has no straight sides or corners.
• A parallelogram (slanted shape) is not a rectangle because its corners are not square corners.
• Any irregular shape without 4 right angles is not a rectangle.

Explanation: A shape is not a rectangle if it does not have exactly 4 sides, 4 square corners, and equal opposite sides.

Given: A square sheet of paper with 4 corners.

Exploration: When we fold one corner of a square inward, the folded corner creates new corners.

Answer: Yes! If we fold all 4 corners of a square sheet inward to the centre, each fold creates new corners. The resulting shape (an octagon or smaller square rotated 45°) can still have 4 corners if folded carefully to meet at the centre, forming a smaller square.

Conclusion: Yes, by folding all 4 corners of a square to the centre, we get a new square shape — the number of corners remains $\mathbf{4}$.

Given: A square is cut along its two diagonals to get 4 right-angled triangles.

Concept: Arrange 3 or 4 triangles in different ways (edge-to-edge) to form different shapes.

With 3 triangles:
By joining 3 right-angled triangles in different arrangements, you can make:

• A larger triangle
• A trapezium (quadrilateral with one pair of parallel sides)
• A parallelogram-like shape

$$\text{Approximately } \mathbf{3} \text{ to } \mathbf{4} \text{ different shapes can be made with 3 triangles.}$$

With 4 triangles:
By joining all 4 triangles, you can make:

• The original square
• A rectangle
• A larger triangle
• A parallelogram

$$\text{Approximately } \mathbf{4} \text{ to } \mathbf{5} \text{ different shapes can be made with 4 triangles.}$$

(Draw and show each arrangement on dot grid paper.)

Given: Several shapes (rectangle, triangle, irregular shapes, etc.) are shown.

Concept: A square corner (right angle) is exactly $90°$ — like the corner of a square or rectangle. We can check using the corner of a sheet of paper or two paper strips.

Method: Place the corner of a square piece of paper at each corner of the shape. If it fits exactly, it is a square corner.

Answer:

• Rectangle: All 4 corners are square corners. Mark all 4.
• Square: All 4 corners are square corners. Mark all 4.
• Triangle (right-angled): One corner is a square corner. Mark that one.
• Irregular shapes: Check each corner with the paper corner test and mark accordingly.

(This is a hands-on marking activity. Use a paper corner to check and mark all square corners in each shape.)

Given: A grid of dots is provided.

Concept: A square has 4 equal sides and 4 square corners. On a dot grid, squares can be made in different sizes and orientations (including tilted squares).

Method:

1. Connect 4 dots to form a square with sides along the grid lines (upright squares).
2. Also try connecting dots diagonally to form tilted squares.

Answer: The number of different squares depends on the dot grid size. On a $4 \times 4$ dot grid, you can make squares of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$, and also tilted squares.

(Count and draw all possible squares on the dot grid provided.)

Given: A picture of a shape (figure not visible in OCR) is shown.

Concept:

• A corner is a point where two sides of a shape meet.
• A square corner is a right angle ($90°$), like the corner of a square or rectangle.

Method: Count all corners of the shape. Then use a paper corner to check which ones are square corners and circle them.

Answer:
(a) Count all the points where sides meet — that gives the number of corners.
(b) Check each corner with a paper square corner; circle those that match exactly.

(The exact answers depend on the figure in the book. Apply the method described above.)

Given: Two matchsticks are available.

Concept: Two matchsticks placed at right angles to each other form square corners.

For two square corners:

• Place two matchsticks in an L-shape (perpendicular to each other). This creates 2 square corners (one on each side of the joint).

For four square corners:

• Place two matchsticks to form a plus (+) sign (one horizontal, one vertical, crossing at the centre). This creates 4 square corners.

Draw:
$$\text{L-shape} \rightarrow 2 \text{ square corners}$$
$$\text{Plus (+) shape} \rightarrow 4 \text{ square corners}$$

(Draw the L-shape and plus-sign in the space provided.)