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Chapter 1 of 14
NCERT Solutions

Shapes Around Us

CBSE · Class 4 · Mathematics

NCERT Solutions for Shapes Around Us — CBSE Class 4 Mathematics.

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Chapter 1: Shapes Around Us

1What parts of the building have you shown in your model (for example, roof, pillars, base, etc.)?Show solution
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.

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2Why did you select these parts?Show solution
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.

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3What shapes will model these parts well?Show solution
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

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4How is your model similar to the picture of the real building?Show solution
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.

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5How is it different from the real building?Show solution
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.

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Discussion-1Discussion: What would happen if you removed one piece of your model? Would the model still look like the original building?Show solution
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.

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Discussion-2In what ways could you make the model even better?Show solution
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.

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Qutub-1Do you think the model looks like the Qutub Minar? What shape would you use if you made a model of the Qutub Minar? Why?Show solution
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.

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Qutub-2How many such shapes will you use?Show solution
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.

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Qutub-3What is common to all of these bricks (clay bricks, stone blocks, wood, concrete blocks, hollow blocks)?Show solution
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.

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Craft-1Make a sphere-like shape with paper strips.Show solution
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).

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Craft-2Use the nets given at the end of the book to make the models shown (Prisms and Pyramids). What shape of face is common to all the prisms? What other shapes do these prisms have? How many such faces each?Show solution
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.

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Craft-3What shape of face is common to all the pyramids? All the triangular faces meet at ___ point. Identify any other shape in each of the pyramids.Show solution
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

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Craft-4Is a cube also a prism?Show solution
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.

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Craft-5What is the difference between a prism and a pyramid? Discuss.Show solution
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.

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3Now try to make the above shapes using straws and plasticine/thread and fill in the table:

| Shapes | Number of faces (F) | Number of corners (V) | Number of edges (E) |
|---|---|---|---|
| Cube/Square Prism | | | |
| Cuboid/Rectangular Prism | | | |
| Triangular Pyramid | | | |
| Square Pyramid | | | |
| Triangular Prism | | | |

Identify any relationship between F, V, and E. Calculate F + V − E in each case. What do you notice?
Show solution
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+812=26 + 8 - 12 = 2
- Cuboid: 6+812=26 + 8 - 12 = 2
- Triangular Pyramid: 4+46=24 + 4 - 6 = 2
- Square Pyramid: 5+58=25 + 5 - 8 = 2
- Triangular Prism: 5+69=25 + 6 - 9 = 2

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

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Sort-1Sort 3D shapes by the number of flat faces. Write their names here.

| 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 | | | | | | |
Show solution
| 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.

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Sort-2Can you construct a 3D shape with 3 flat faces?Show solution
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.

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Sort-3Now sort 3D shapes by the number of straight edges.

| Number of edges | 6 straight edges | 8 straight edges | 9 straight edges | 12 straight edges |
|---|---|---|---|---|
| Name of the shape | | | | |
Show solution
| 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 |

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LetUsObserve-1aTake a die. The face numbered 6 is opposite to the face numbered 1. What is the face opposite to the face numbered 2?Show solution
On a standard die, opposite faces always add up to 7.

Face numbered 2: 72=57 - 2 = 5

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

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LetUsObserve-1bWhat is the face opposite to the face numbered 3?Show solution
On a standard die, opposite faces add up to 7.

Face numbered 3: 73=47 - 3 = 4

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

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LetUsObserve-1cWhat is the face opposite to the face numbered 4?Show solution
On a standard die, opposite faces add up to 7.

Face numbered 4: 74=37 - 4 = 3

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

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LetUsObserve-2aWhich faces have common edges with the face numbered 1?Show solution
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.)

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LetUsObserve-2bWhich face has no common edge with the face numbered 1?Show solution
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.

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LetUsObserve-3aLook at three different views of the same cube. What colour is the face that is opposite to the red face?Show solution
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.)

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LetUsObserve-3bWhat colour is the face that is opposite to the yellow face?Show solution
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.)

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BorderActivityFollow these instructions for the shapes along the border:
1. Colour all shapes with a rectangular face in red.
2. Draw a smiley on shapes with a triangular face.
3. Draw a star on shapes with a curved face.
4. Colour all shapes with no corner in blue.
5. Circle the shapes that have the same opposite faces.
Show solution
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).

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Sorting3DWrite the names of 3D shapes in the correct places in the Venn diagram (sorting by properties like 'has a vertex/apex' and 'has a curved face').Show solution
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.

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Sorting3D-2In which circle did you write triangular prism and rectangular pyramid?Show solution
- 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.

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Sorting3D-3Using circles, sort shapes into the categories 'Shapes with curved faces' and 'Shapes with flat faces'.Show solution
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.

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CubeTowersHow many cubes are there in each of these cube towers? (Refer to the figures in the textbook.)Show solution
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.

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Match-1aMatch the pictures to the descriptions and name the shapes.
a) I have 5 faces and 5 corners. I have 8 edges. 1 of my faces is a square and 4 of my faces are triangles.
Show solution
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+VE=5+58=2F + V - E = 5 + 5 - 8 = 2

Answer: Square Pyramid

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Match-1bb) I have 1 flat face, 1 curved face, and 1 edge.Show solution
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

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Match-1cc) I have 1 curved face. I have no edges or corners.Show solution
Given: 1 curved face, no edges, no corners.

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

Answer: Sphere

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Match-1dd) I have 2 flat faces, 1 curved face, and 2 edges. I have no corners.Show solution
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

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Match-1ee) I have 5 faces, 6 corners, and 9 edges, and 2 of my faces are triangles.Show solution
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+VE=5+69=2F + V - E = 5 + 6 - 9 = 2

Answer: Triangular Prism

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Match-1ff) I have 6 faces, 12 edges, and 8 corners.Show solution
Given: 6 faces, 12 edges, 8 corners.

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

Verification: F+VE=6+812=2F + V - E = 6 + 8 - 12 = 2

Answer: Cube or Cuboid

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Match-2Each one is different. How? Discuss. (Referring to figures of similar-looking shapes)Show solution
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.

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Match-3Match the following nets to the appropriate solids given below. (Nets 1 through 4 matched to solids a, b, c, d)Show solution
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.

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Match-4Which of these nets (A, B, C) can be folded to make the solid given?Show solution
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.)

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Match-5Nitesh cuts up a net on the folds. Here are its pieces (1 square and 4 triangles). Which solid has the above pieces in its net? (Options: a, b, c, d)Show solution
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.

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BoatAnglesThere are 7 angles in the house drawing. How many angles are there in the boat drawing?Show solution
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.)

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LetUsDo-1Mark the angles in the following pictures (a, b, c, d, e).Show solution
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.

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LetUsDo-2Where do you see angles in the classroom? Give a few examples.Show solution
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

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RightAngles-1Identify the angles that you think are right angles and circle them in the dot grid. Check using your right angle checker.
RightAngles-2Check for right angles in a book, window, and any other object. Write the names of objects where you find right angles.
LetUsDo-DrawRightDraw some right angles on the dot grid.
AcuteObtuse-1Name some objects from your classroom which have an acute angle.
AcuteObtuse-2Name some objects from your classroom which have an obtuse angle.
AcuteObtuse-3Identify all angles in the following letters (shown in the figures — typically letters like A, V, W, etc.).
LetUsDo-DrawAcuteDraw some acute angles on the top grid. Draw a line to make an acute angle using each given line in the bottom grid. Draw some obtuse angles on the top grid. Draw a line to make an obtuse angle using each given line in the bottom grid.
LetUsDo-2-MarkAnglesIn the figures given below, mark the acute angles in red, right angles in green, and obtuse angles in blue.
StrawShapes-1Does the shape of the triangle change if we gently push one of its sides? Yes/No
StrawShapes-2What kinds of angles does a triangle have?
StrawShapes-3What kinds of angles do you see in the rectangle?
StrawShapes-4Does the shape of the rectangle change if we gently push one of its sides? Yes/No
StrawShapes-5What has happened to the angles of the new shape? Are they still right angles? What types of angles have been formed?
StrawShapes-6Similarly, push one side of a square. Are they still right angles? What types of angles have been formed?
StrawShapes-7How are the angles of triangles and rectangles similar or different?
DotGrid-DiscussUse the dot grid to draw several three- and four-sided shapes. Discuss: How many shapes have you made with (a) 1 right angle, (b) 2 right angles, (c) 3 right angles, (d) all right angles?
FourSidedShapesHere are some 4-sided shapes. In what ways are rectangle and square different from these shapes?
Pentagon-1Are the angles of a regular pentagon right angles?
Pentagon-2Does the shape of the pentagon change if we gently push one of its sides? Yes/No. How does this change the angles?
CircleStraws-1Can you make a circle using straws? The lengths of the straws in the picture are ___ (Equal/Unequal). What will happen if we take straws of unequal lengths?
CircleActivity-1Amazing Circles: The length of all the creases are ___ (Equal/Unequal).
CircleActivity-2These creases are called ___ of the circle. Do all the diameters pass through the centre?
CircleActivity-3Discuss if there is any relationship between the radius and the diameter of a circle.
LetUsDo-DiameterFold the circular paper in half. Fold this half again in half. The length of the diameter is ___ (half/double) of the length of radius.
Wheels-1All wheels look like ___
Wheels-2Name the wheel with the (1) longest radius, (2) shortest radius, (3) longest diameter, (4) shortest diameter.
Puzzling-1Identify the hidden shapes and write their names.
Puzzling-2Draw 2 lines to divide the triangle into 1 square and 2 triangles.
Puzzling-3Draw 2 lines to divide the square into 3 triangles.
Puzzling-4Draw lines to show the cuts needed on the shapes in the left column to get the smaller shapes on the right. (A), (B), (C)
CardGameSort the 2D-shape cards into three groups according to their sides. Draw the sorted shapes and explain why you sorted them this way.
LetUsTry-1Squiggly, the spider, makes triangular webs. How many triangles are in her web? Can she begin at point A and reach back to the same point without walking on any wall more than once? Trace and show Squiggly's path.
LetUsTry-1bWiggly made a web using rectangles. How many rectangles can you see in his web? Can he begin at point A and leave from point B without walking on any wall more than once?
LetUsTry-2Use 5 matchsticks to make 2 triangles. Then draw it in the space provided.
LetUsTry-3Move two of these matchsticks to form 4 triangles.
LetUsTry-4Remove 4 of these matchsticks to leave only 3 triangles.
LetUsTry-5aModel challenge. Can you make a model of solid shapes which has (a) 12 straws and 8 clay balls?
LetUsTry-5b(b) 9 straws and 6 clay balls?
LetUsTry-5c(c) 15 straws and 10 clay balls?
LetUsTry-5d(d) 10 straws and 6 clay balls?
LetUsTry-6Classify these shapes based on the number of angles.
3 angles | 4 angles | 5 angles
What relation do you notice between the number of sides and the number of angles?
LetUsTry-7Draw a 2D shape that has less than 5 angles. Draw a 2D shape with more than 5 angles.
LetUsTry-8Mark the right angles and write the number of right angles in each figure. Which of the above shapes have only right angles?
LetUsTry-9Observe the following shapes. Identify the shape that has:
- 2 right angles, 1 acute, and 1 obtuse angle
- 1 right, 2 obtuse, and 1 acute angle
- 2 obtuse, and 2 acute angles
- 4 right angles

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Frequently Asked Questions

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Shapes Around Us covers several key topics that are frequently asked in CBSE Class 4 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Shapes Around Us — CBSE Class 4 Mathematics?
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