Shapes Around Us

Given: We are making a model of a building (like India Gate) using wooden blocks.

Answer (sample): In my model I have shown the following parts:
- Base (the flat bottom platform)
- Pillars (tall rectangular columns on the sides)
- Arch (the curved gateway in the middle)
- Top slab (the flat roof portion)

Note: Answers may vary depending on the building chosen and the blocks available.

I selected these parts because they are the most important and recognisable features of the building. The base, pillars, and arch together give the model its overall shape and make it look like the real building. Without these key parts, the model would not be recognisable.

The shapes that model the parts well are:
- Base → Cuboid (rectangular block)
- Pillars → Cuboids (tall, thin rectangular blocks)
- Arch → A curved or semi-circular block
- Top slab → Cuboid (flat rectangular block)
- Dome or roof → Hemisphere or cone shape

My model is similar to the real building in the following ways:
- It has the same basic structure: a base, pillars, and a top.
- The overall outline and silhouette looks like the real building.
- The relative positions of the parts (base at the bottom, pillars in the middle, roof at the top) are the same.

My model is different from the real building in the following ways:
- The model is much smaller in size.
- The model does not have fine carvings, decorations, or inscriptions.
- The materials used (wooden blocks) are different from the real materials (stone, concrete).
- The colours and textures are different.
- Some smaller details like windows, inscriptions, and lighting cannot be shown.

If one important piece (like a pillar or the arch) is removed, the model will no longer look like the original building because the key recognisable feature will be missing. However, if a small or less important piece is removed, the model may still roughly resemble the building.

We could make the model better by:
- Using more blocks to add finer details.
- Painting the model to match the colour of the real building.
- Adding carvings or decorations using clay.
- Using curved blocks to better represent arches and domes.
- Making the model to a proper scale so proportions are correct.

The Qutub Minar is a tall, tapering tower. To model it, I would use a cylinder or a cone shape (or a combination of cylinders of decreasing size stacked on top of each other). I would choose this shape because the Qutub Minar is round and tall, and a cylinder best represents its circular cross-section and height.

Since the Qutub Minar has 5 storeys, I would use 5 cylinders of slightly decreasing size (each one a little narrower than the one below) stacked on top of each other to show the tapering effect.

What is common to all these bricks is that they are all cuboid (box) shaped — they have flat faces, straight edges, and right-angle corners. This shape makes them easy to stack and build with.

Activity: Take several strips of paper. Staple or glue them together at the top and bottom, spreading them out evenly around the centre to form a round, sphere-like shape. The strips curve outward in the middle, giving the appearance of a sphere (globe shape).

Prisms:
- The shape of face common to all prisms is a rectangle (the lateral/side faces are rectangles).
- The other shapes are the two end faces (bases), which match the name of the prism:
- Triangular Prism → 2 triangular faces
- Square Prism (Cube) → 2 square faces
- Hexagonal Prism → 2 hexagonal faces
- Each prism has 2 such end faces.

Pyramids:
- The shape of face common to all pyramids is a triangle (the lateral faces are triangles).
- All the triangular faces meet at one (1) point (called the apex).
- The other shape in each pyramid is the base:
- Triangular Pyramid → triangular base
- Pentagonal Pyramid → pentagonal base

Yes, a cube is also a prism — specifically, it is a square prism where all faces are squares (equal squares). It has two square bases and four square lateral faces, which satisfies the definition of a prism.

Prism: Has two identical and parallel bases (top and bottom faces) connected by rectangular lateral faces. Example: Triangular prism, cuboid.

Pyramid: Has only one base and all other faces are triangles that meet at a single point called the apex. Example: Square pyramid, triangular pyramid.

Key difference: A prism has two bases; a pyramid has only one base and comes to a point at the top.

Filled Table:

| Shapes | F (Faces) | V (Corners) | E (Edges) |
|---|---|---|---|
| Cube/Square Prism | 6 | 8 | 12 |
| Cuboid/Rectangular Prism | 6 | 8 | 12 |
| Triangular Pyramid | 4 | 4 | 6 |
| Square Pyramid | 5 | 5 | 8 |
| Triangular Prism | 5 | 6 | 9 |

Calculating F + V − E:
- Cube: $6 + 8 - 12 = 2$
- Cuboid: $6 + 8 - 12 = 2$
- Triangular Pyramid: $4 + 4 - 6 = 2$
- Square Pyramid: $5 + 5 - 8 = 2$
- Triangular Prism: $5 + 6 - 9 = 2$

Observation: In every case, $F + V - E = 2$. This is known as Euler's Formula for 3D shapes.

| Number of faces | 1 flat face | 2 flat faces | 4 flat faces | 5 flat faces | 6 flat faces | 8 flat faces |
|---|---|---|---|---|---|---|
| Name of the shape | Cone | Cylinder | Triangular Pyramid (Tetrahedron) | Triangular Prism / Square Pyramid | Cube / Cuboid | Hexagonal Prism |

Note: A cone has 1 flat (circular) face and 1 curved face. A cylinder has 2 flat (circular) faces and 1 curved face.

Yes, a triangular prism has 5 faces in total — but if we think of a shape with exactly 3 flat faces, we can imagine a shape like a wedge (a prism with a very thin triangular cross-section). However, among standard 3D shapes, it is difficult to have exactly 3 flat faces without any curved face. A shape with 2 triangular faces and 1 rectangular face would be a very flat triangular prism. In practice, constructing such a shape is possible but it is not a common standard solid.

| Number of edges | 6 straight edges | 8 straight edges | 9 straight edges | 12 straight edges |
|---|---|---|---|---|
| Name of the shape | Triangular Pyramid (Tetrahedron) | Square Pyramid | Triangular Prism | Cube / Cuboid |

On a standard die, opposite faces always add up to 7.

Face numbered 2: $7 - 2 = 5$

The face opposite to face numbered 2 is face numbered 5.

On a standard die, opposite faces add up to 7.

Face numbered 3: $7 - 3 = 4$

The face opposite to face numbered 3 is face numbered 4.

On a standard die, opposite faces add up to 7.

Face numbered 4: $7 - 4 = 3$

The face opposite to face numbered 4 is face numbered 3.

The face numbered 1 on a standard die is surrounded by four faces that share an edge with it. These are the faces numbered 2, 3, 4, and 5.

(Face 6 is directly opposite to face 1 and shares no edge with it.)

The face that is directly opposite to face 1 has no common edge with it.

The face numbered 6 has no common edge with face numbered 1.

By carefully observing the three views of the cube shown (the images show different orientations), we can deduce which colour is opposite to red.

Note: Since the actual images cannot be seen, the standard answer given in the NCERT textbook is:
The face opposite to the red face is blue.

(Students should verify this by looking at the three views provided in their textbook.)

By observing the three views of the cube:

The face opposite to the yellow face is green.

(Students should verify this by looking at the three views provided in their textbook.)

This is a hands-on activity. Here are the guidelines:

1. Colour red → Cuboid, Cube, Triangular Prism, Square Prism, Hexagonal Prism (all have rectangular faces).
2. Draw a smiley → Triangular Prism, Triangular Pyramid, Square Pyramid (have triangular faces).
3. Draw a star → Cylinder, Cone, Sphere (have curved faces).
4. Colour blue → Sphere (has no corners or edges); Cylinder and Cone have no corners but have edges — Sphere alone has no corner.
5. Circle → Cube, Cuboid (opposite faces are identical/same shape and size).

This is a Venn-diagram sorting activity. General guidance:

- Shapes with only flat faces (no curved face): Cube, Cuboid, Triangular Prism, Square Prism, Triangular Pyramid, Square Pyramid, Hexagonal Prism.
- Shapes with only curved faces: Sphere.
- Shapes with both flat and curved faces: Cylinder (2 flat + 1 curved), Cone (1 flat + 1 curved).

Triangular prism goes in the 'prisms' circle. Rectangular pyramid goes in the 'pyramids' circle. Neither belongs to the overlapping region since a prism is not a pyramid.

- Triangular prism was written in the Prisms circle.
- Rectangular pyramid was written in the Pyramids circle.

Neither shape belongs to the overlapping (intersection) region because a shape cannot be both a prism and a pyramid at the same time.

Shapes with curved faces only: Sphere.

Shapes with flat faces only: Cube, Cuboid, Triangular Prism, Square Prism, Hexagonal Prism, Triangular Pyramid, Square Pyramid.

Shapes with BOTH curved and flat faces (overlapping region): Cylinder, Cone.

Draw two overlapping circles. Place Sphere in the 'curved only' part, Cube/Cuboid/Prisms/Pyramids in the 'flat only' part, and Cylinder/Cone in the overlapping middle part.

This is an observation-based activity using the figures in the textbook. General method:

Step 1: Count the cubes visible in each layer.
Step 2: Add the cubes in all layers.

Typical answers (based on standard NCERT figures):
- Tower 1: 3 cubes (3 cubes stacked vertically)
- Tower 2: 6 cubes (2 layers of 3 cubes each)

Students should count carefully from the figures in their textbook, including any hidden cubes in the back.

Given: 5 faces, 5 corners, 8 edges; 1 square face and 4 triangular faces.

Concept: A shape with a square base and 4 triangular faces meeting at a point is a Square Pyramid.

Verification using Euler's Formula: $F + V - E = 5 + 5 - 8 = 2$ ✓

Answer: Square Pyramid

Given: 1 flat face, 1 curved face, 1 edge, and it comes to a point (apex).

Concept: A cone has a circular flat base (1 flat face), a curved lateral surface (1 curved face), and one circular edge where the flat and curved faces meet.

Answer: Cone

Given: 1 curved face, no edges, no corners.

Concept: A sphere has only one curved surface with no edges and no corners.

Answer: Sphere

Given: 2 flat faces, 1 curved face, 2 edges, no corners.

Concept: A cylinder has 2 circular flat faces (top and bottom), 1 curved lateral face, and 2 circular edges. It has no corners (vertices).

Answer: Cylinder

Given: 5 faces, 6 corners, 9 edges; 2 triangular faces.

Concept: A triangular prism has 2 triangular faces (the two bases) and 3 rectangular faces, giving 5 faces total, 6 vertices, and 9 edges.

Verification: $F + V - E = 5 + 6 - 9 = 2$ ✓

Answer: Triangular Prism

Given: 6 faces, 12 edges, 8 corners.

Concept: A cube (or cuboid) has 6 faces, 12 edges, and 8 corners.

Verification: $F + V - E = 6 + 8 - 12 = 2$ ✓

Answer: Cube or Cuboid

The shapes may look similar at first glance but differ in:
- Number of faces: e.g., a cube has 6 equal square faces while a cuboid has 6 rectangular faces (not all equal).
- Size of faces: All faces of a cube are equal squares; a cuboid's faces are rectangles of different sizes.
- Angles: Some shapes have right angles, others do not.
- Symmetry: Some shapes are more symmetric than others.

Discuss with your teacher and classmates how each shape in the figure is unique.

This is a visual matching activity. The general approach is:

Step 1: Count the number of faces in the net.
Step 2: Identify the shapes of the faces.
Step 3: Match to the solid that has the same faces.

General matching rules:
- A net with 6 squares → Cube
- A net with 2 triangles + 3 rectangles → Triangular Prism
- A net with 1 square + 4 triangles → Square Pyramid
- A net with 2 circles + 1 rectangle → Cylinder

Students should fold each net mentally (or physically) to identify the solid it forms, then match accordingly.

Method: To check if a net can be folded into a given solid:
1. Count the faces in the net and compare with the solid.
2. Check that opposite faces will align correctly when folded.
3. Ensure no faces overlap when folded.

For a cube (6 square faces): Only nets where the 6 squares are arranged so that no two squares will overlap when folded are valid.

Answer: Net B can typically be folded to make the given solid. (Students should verify by cutting and folding the nets provided in their textbook.)

Given pieces: The pieces consist of 1 square and 4 triangles.

Analysis:
- 1 square base + 4 triangular faces = Square Pyramid
- Total faces = 5, corners = 5, edges = 8

Answer: Option b — Square Pyramid

The net with 1 square and 4 triangles folds into a Square Pyramid.

Method: Count every place where two straight lines meet in the boat drawing. Each such meeting point forms an angle.

By carefully examining a typical boat outline drawing (a shape with a pointed front, flat bottom, and slanted sides):

Answer: The boat drawing has approximately 5 angles.

(The exact number depends on the specific drawing in the textbook. Students should count each corner/vertex in the boat figure.)

Method: An angle is formed wherever two straight lines (or sides) meet at a point. Mark a small arc at each such meeting point to indicate the angle.

- (a) Mark angles at each corner of the shape shown.
- (b) Mark angles at each corner of the shape shown.
- (c) Mark angles at each corner of the shape shown.
- (d) Mark angles at each corner of the shape shown.
- (e) Mark angles at each corner of the shape shown.

This is a hands-on activity. Students should look at each figure and place a small curved mark (arc) at every corner where two sides meet.

We can see angles in many places in the classroom:

1. Corners of the blackboard — right angles (90°)
2. Corners of books and notebooks — right angles
3. Corners of the door and windows — right angles
4. Legs of a chair or table meeting the floor — right angles
5. The hands of a clock — form different angles at different times
6. The corner of a set square — acute and right angles
7. The tip of a sharpened pencil — acute angle

Method:
1. Make a right angle checker by folding a piece of paper twice to get a perfect right angle (90°).
2. Place the checker at each angle in the dot grid figures.
3. If the angle matches the checker exactly, it is a right angle — circle it.
4. If the angle is smaller, it is acute; if larger, it is obtuse.

This is a hands-on activity. Students should use their paper right-angle checker to test each angle in the figures on the dot grid.

Objects that have right angles:
1. Book — all four corners are right angles
2. Window — all four corners are right angles
3. Door — all four corners are right angles
4. Blackboard — all four corners are right angles
5. Table top — all four corners are right angles
6. Notebook — all four corners are right angles
7. Floor tiles — all four corners are right angles

Method to draw a right angle on a dot grid:
1. Choose a dot as the vertex (corner point).
2. Draw one line going horizontally to the right (along a row of dots).
3. Draw another line going vertically upward (along a column of dots).
4. The angle between these two lines is exactly 90° — a right angle.
5. Mark a small square at the corner to show it is a right angle.

Repeat this at different positions on the dot grid to draw several right angles.

Objects in the classroom with acute angles:
1. The tip of a sharpened pencil — the pointed end forms an acute angle.
2. The blade of a pair of scissors when open slightly.
3. The corner of a triangular set square (the 30° and 60° angles).
4. The pointed tip of a compass.
5. The corner of a folded paper when folded at a sharp angle.

Objects in the classroom with obtuse angles:
1. An open book placed flat — the spine forms an obtuse angle.
2. The hands of a clock at certain times (e.g., at 10:10, the angle between hands is obtuse).
3. A door partially open — the angle between the door and the wall can be obtuse.
4. The back of a reclining chair — forms an obtuse angle with the seat.
5. A pair of scissors opened wide — the angle between the blades becomes obtuse.

Method: Look at each letter and identify where two lines meet.

- Letter A: Has 2 acute angles at the top (where the two slanted lines meet) and 2 obtuse angles at the base corners. The crossbar creates additional angles.
- Letter V: Has 1 acute angle at the bottom point.
- Letter W: Has multiple acute and obtuse angles at the peaks and valleys.
- Letter Z: Has 2 acute angles and 2 obtuse angles.
- Letter N: Has 2 right angles and 2 acute angles.

Students should examine the specific letters shown in their textbook and mark each angle as acute, right, or obtuse.

Drawing Acute Angles:
- An acute angle is less than a right angle (less than 90°).
- On the dot grid, draw two lines from a common point so that the opening between them is smaller than a right angle.
- Example: Draw one line going right, and another going diagonally up-right at a small opening.

Drawing Obtuse Angles:
- An obtuse angle is more than a right angle (more than 90° but less than 180°).
- On the dot grid, draw two lines from a common point so that the opening between them is larger than a right angle.
- Example: Draw one line going right, and another going diagonally up-left so the angle between them is wide.

This is a hands-on drawing activity.

Method:
1. Use your right-angle checker (folded paper) to test each angle in the figures.
2. If the angle matches the checker → Right angle → mark in green.
3. If the angle is smaller than the checker → Acute angle → mark in red.
4. If the angle is larger than the checker → Obtuse angle → mark in blue.

This is a hands-on colouring activity. Students should apply this method to each figure in their textbook.

Answer: No

Explanation: A triangle is a rigid shape. When you push one of its sides, the shape does not change because the three sides lock each other in place. This is why triangles are used in bridges and buildings for strength and stability.

A triangle can have:
- Acute angles (less than 90°) — in an acute triangle, all three angles are acute.
- One right angle (exactly 90°) — in a right-angled triangle.
- One obtuse angle (more than 90°) — in an obtuse triangle.

In a triangle made with straws of different sizes, the angles will typically be a mix of acute and obtuse angles (or one right angle).

A rectangle has 4 right angles (each angle is exactly 90°). All four corners of a rectangle are right angles.

Answer: Yes

Explanation: Unlike a triangle, a rectangle (or any quadrilateral) is not rigid. When you push one of its sides, the shape changes — the right angles become acute and obtuse angles, and the rectangle becomes a parallelogram shape.

When the rectangle is pushed:
- The right angles are no longer right angles.
- Two of the angles become acute angles (less than 90°).
- The other two angles become obtuse angles (more than 90°).
- The shape is now a parallelogram.

When a square is pushed:
- The right angles are no longer right angles.
- Two angles become acute and two become obtuse.
- The shape changes from a square to a rhombus (a parallelogram with equal sides).

Similarity: Both triangles and rectangles have angles formed where sides meet.

Differences:
- A rectangle always has 4 right angles; a triangle has 3 angles which may be acute, right, or obtuse.
- When pushed, a rectangle's angles change (it is not rigid), but a triangle's angles do not change (it is rigid).
- A triangle has 3 angles that always add up to 180°; a rectangle has 4 angles that add up to 360°.

This is a drawing and discussion activity. Key observations:

(a) 1 right angle: A triangle with one 90° corner (right-angled triangle) has exactly 1 right angle.

(b) 2 right angles: A right trapezium (a four-sided shape with exactly 2 right angles) is an example.

(c) 3 right angles: Interestingly, a four-sided shape cannot have exactly 3 right angles — if three angles are 90° each ($3 \times 90° = 270°$), the fourth must also be $360° - 270° = 90°$. So a four-sided shape with 3 right angles automatically has 4 right angles. For a triangle, having 3 right angles is impossible since 3 \times 90° = 270° > 180°.

(d) All right angles: A rectangle or square has all 4 right angles.

A rectangle and square are special 4-sided shapes (quadrilaterals) that are different from other 4-sided shapes because:

1. All angles are right angles (90°) — other quadrilaterals like parallelograms, trapeziums, and rhombuses do not have all right angles.
2. Opposite sides are equal and parallel — rectangles have two pairs of equal opposite sides; squares have all four sides equal.
3. Squares are a special rectangle where all four sides are equal.
4. Other 4-sided shapes may have unequal sides, unequal angles, or no right angles at all.

No, the angles of a regular pentagon are not right angles.

Each interior angle of a regular pentagon = $\frac{(5-2) \times 180°}{5} = \frac{3 \times 180°}{5} = \frac{540°}{5} = 108°$

Since 108° > 90°, each angle of a regular pentagon is an obtuse angle, not a right angle.

Answer: Yes

When one side of a pentagon (made with straws) is pushed:
- The shape changes because a pentagon, like a rectangle, is not rigid.
- The angles that were obtuse may become more acute or more obtuse depending on the direction of the push.
- The shape distorts and is no longer a regular pentagon.

Answer: Equal

To make a circle using straws, all the straws must be of equal length and arranged so that they radiate from a central point at equal angles.

If straws of unequal lengths are used: The shape formed will not be a circle — it will be an irregular, uneven shape (not perfectly round). The boundary will be closer to the centre in some places and farther in others.

Answer: Equal

When a circular piece of paper is folded in half in different ways, each fold creates a crease that passes through the centre of the circle. All such creases (diameters) are equal in length because the diameter of a circle is always the same regardless of the direction of the fold.

These creases are called diameters of the circle.

Yes, all the diameters pass through the centre of the circle. This is a fundamental property of a circle — every diameter must pass through the centre point.

Relationship between Radius and Diameter:

The diameter is the length of the crease from one edge of the circle to the other, passing through the centre.

The radius is the length from the centre to the edge of the circle.

Since the diameter passes through the centre and goes from one side to the other:
$$\text{Diameter} = 2 \times \text{Radius}$$
$$\text{Radius} = \frac{\text{Diameter}}{2}$$

The diameter is always double the radius, and the radius is always half the diameter.

Answer: double

Explanation:
When you fold the circle in half, the crease is the diameter.
When you fold that half again, the crease is the radius (half of the diameter).

Therefore: Diameter = 2 × Radius

The diameter is double the length of the radius.

All wheels look like circles.

Since the actual wheel images cannot be seen, the method to answer is:

Concept: The larger the wheel (bigger circle), the longer its radius and diameter. The smaller the wheel, the shorter its radius and diameter.

- (1) Longest radius → The largest wheel in the picture.
- (2) Shortest radius → The smallest wheel in the picture.
- (3) Longest diameter → The largest wheel (same as longest radius, since diameter = 2 × radius).
- (4) Shortest diameter → The smallest wheel (same as shortest radius).

Students should identify the largest and smallest wheels from the figures in their textbook.

This is a visual identification activity. In the given figure, look carefully for shapes hidden within the larger design.

Common hidden shapes found in such puzzles:
- Triangles (small and large)
- Squares
- Rectangles
- Parallelograms
- Rhombuses

Students should examine the figure carefully and list all the shapes they can find, including overlapping and combined shapes.

Method:
1. Start with a large right-angled triangle (or an equilateral triangle as shown).
2. Draw a vertical line from a point on the top side straight down to the base — this creates a square in the middle.
3. Draw a horizontal line to complete the square.

Result: The two lines divide the triangle into:
- 1 square (in the centre/corner)
- 2 smaller triangles (on either side of the square)

This is a drawing activity. Students should try different positions for the lines to achieve the required division.

Method:
1. Start with a square.
2. Draw one diagonal line from one corner to the opposite corner — this divides the square into 2 triangles.
3. Draw another line from one corner to the midpoint of the opposite side — this divides one of the triangles into 2 smaller triangles.

Result: The square is now divided into 3 triangles.

Alternatively: Draw a line from one corner to a point on the opposite side (not the midpoint), then draw another line from a different corner to create 3 triangles.

This is a visual cutting/dividing activity.

(A) Identify the large shape on the left and the smaller shapes on the right. Draw lines on the large shape to show how it can be cut to produce the smaller shapes.

(B) Similarly, draw the cut lines on the second shape.

(C) Similarly, draw the cut lines on the third shape.

General approach: Look at the smaller shapes and figure out how they fit together to make the larger shape. The cut lines are the boundaries between the smaller shapes.

Students should work with the specific figures in their textbook.

Sorting 2D shapes by number of sides:

Group 1 — 3 sides (Triangles):
- Equilateral triangle, right-angled triangle, isosceles triangle
- *Reason:* All have exactly 3 sides and 3 angles.

Group 2 — 4 sides (Quadrilaterals):
- Square, rectangle, rhombus, parallelogram, trapezium
- *Reason:* All have exactly 4 sides and 4 angles.

Group 3 — 5 or more sides (Polygons):
- Pentagon (5 sides), hexagon (6 sides), octagon (8 sides)
- *Reason:* All have 5 or more sides.

Why sorted this way: Grouping by number of sides is a clear and logical way to classify 2D shapes, as the number of sides determines many other properties of the shape.

Number of triangles in the web:
By examining a typical triangular web pattern (a triangle divided into smaller triangles), count:
- Small triangles
- Medium triangles (made of 2 small triangles)
- Large triangle (the whole web)

A typical answer for a web divided into 4 small triangles: Total = 4 + 0 + 1 = 5 triangles (or as per the specific figure).

Can Squiggly walk all walls without repeating?
This is an Euler path problem. A path that starts and ends at the same point without repeating any edge is possible only if every vertex has an even number of edges.

For a simple triangular web, check the degree of each vertex. If all vertices have even degree → Yes, she can do it. If any vertex has odd degree → No, she cannot return to the same point without repeating a wall.

Students should trace the path on the figure in their textbook.

Number of rectangles:
In a rectangular web (a grid of rectangles), count:
- Individual small rectangles
- Rectangles made of 2 small rectangles
- Rectangles made of 4 small rectangles, etc.

For a 2×2 grid: $4 + 2 + 2 + 1 = 9$ rectangles total.

Can Wiggly walk from A to B without repeating?
This is an Euler path problem. A path from A to B (different start and end points) without repeating edges is possible if exactly 2 vertices have an odd number of edges (and those 2 vertices are A and B).

Students should check the degrees of all vertices in the figure and trace the path if possible.

Solution:
To make 2 triangles using only 5 matchsticks, the two triangles must share one side (one matchstick is common to both triangles).

Arrangement:
- Place 3 matchsticks to form one triangle.
- Use 2 more matchsticks along with one side of the first triangle to form the second triangle.

Drawing: Draw two triangles that share a common side, like a diamond/rhombus shape divided by a diagonal, using 5 matchsticks total:
- Bottom triangle: 3 matchsticks
- Top triangle: shares the top side of the bottom triangle + 2 new matchsticks

This gives 2 triangles with $3 + 2 = 5$ matchsticks.

Given: A figure made of matchsticks (typically showing 3 triangles arranged in a row or pattern).

Solution: Move 2 matchsticks to rearrange the figure so that 4 triangles are formed.

Typical solution: Take the figure of 3 triangles in a row (using 7 matchsticks). Move 2 matchsticks from the outer triangles to create a central arrangement where 4 triangles are visible (including overlapping triangles).

Students should experiment with the specific matchstick figure in their textbook by physically moving 2 matchsticks and counting the resulting triangles.

Given: A figure made of matchsticks showing more than 3 triangles.

Solution: Remove 4 matchsticks such that exactly 3 triangles remain.

Strategy:
1. Count the current number of triangles.
2. Identify which matchsticks, when removed, will eliminate the extra triangles while keeping exactly 3.
3. Remove matchsticks that are shared between triangles you want to eliminate.

Students should work with the specific figure in their textbook and try removing different combinations of 4 matchsticks until exactly 3 triangles remain.

Given: 12 straws (edges) and 8 clay balls (corners/vertices).

Using Euler's formula: $F + V - E = 2$
$F + 8 - 12 = 2$
$F = 6$

Shape with 6 faces, 8 vertices, 12 edges = Cube or Cuboid

Answer: Make a Cube or Cuboid using 12 straws as edges and 8 clay balls as corners.

Given: 9 straws (edges) and 6 clay balls (vertices).

Using Euler's formula: $F + V - E = 2$
$F + 6 - 9 = 2$
$F = 5$

Shape with 5 faces, 6 vertices, 9 edges = Triangular Prism

Answer: Make a Triangular Prism using 9 straws as edges and 6 clay balls as corners.

Given: 15 straws (edges) and 10 clay balls (vertices).

Using Euler's formula: $F + V - E = 2$
$F + 10 - 15 = 2$
$F = 7$

Shape with 7 faces, 10 vertices, 15 edges = Pentagonal Prism

Answer: Make a Pentagonal Prism using 15 straws as edges and 10 clay balls as corners.

Given: 10 straws (edges) and 6 clay balls (vertices).

Using Euler's formula: $F + V - E = 2$
$F + 6 - 10 = 2$
$F = 6$

Shape with 6 faces, 6 vertices, 10 edges = Pentagonal Pyramid

Verification: A pentagonal pyramid has 1 pentagonal base + 5 triangular faces = 6 faces, 5 + 1 = 6 vertices, 5 + 5 = 10 edges. ✓

Answer: Make a Pentagonal Pyramid using 10 straws as edges and 6 clay balls as corners.

Classification:

| 3 angles | 4 angles | 5 angles |
|---|---|---|
| Triangle | Square, Rectangle, Rhombus, Parallelogram, Trapezium | Pentagon |

Relationship between number of sides and number of angles:

The number of sides of a shape is always equal to the number of angles.

- A triangle has 3 sides and 3 angles.
- A quadrilateral has 4 sides and 4 angles.
- A pentagon has 5 sides and 5 angles.

Rule: Number of sides = Number of angles (for any polygon).

Shape with less than 5 angles (e.g., 3 angles):
Draw a triangle — it has 3 sides and 3 angles.

$$\triangle$$

Shape with more than 5 angles (e.g., 6 angles):
Draw a hexagon — it has 6 sides and 6 angles.

Students should draw these shapes in the spaces provided in their textbook. A triangle can be drawn by connecting 3 dots on the dot grid. A hexagon can be drawn by connecting 6 dots.

Method: Use a right-angle checker (folded paper) to test each angle in the figures.

Typical answers for standard figures:
- Square: 4 right angles — mark all 4 corners with a small square symbol.
- Rectangle: 4 right angles — mark all 4 corners.
- Right-angled triangle: 1 right angle — mark the 90° corner.
- L-shape: 6 right angles — mark all 6 corners.
- Irregular quadrilateral: 0 or 1 or 2 right angles depending on the shape.

Shapes with ONLY right angles: Square and Rectangle (all their angles are right angles).

Students should apply the right-angle checker to each specific figure in their textbook.

Method: Examine each shape in the figure and classify its angles.

Answers (based on standard NCERT figures):

- 2 right angles, 1 acute, and 1 obtuse angle → A right trapezium (a trapezium with 2 right angles, one acute, and one obtuse angle).

- 1 right, 2 obtuse, and 1 acute angle → An irregular quadrilateral with these specific angle types.

- 2 obtuse and 2 acute angles → A parallelogram (opposite angles are equal; two are acute and two are obtuse).

- 4 right angles → A rectangle or square (all four angles are 90°).

Students should match these descriptions to the specific shapes shown in their textbook by measuring/checking each angle.