Shapes Around Us — NCERT Solutions

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:

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.