3D Shapes — Boxes, Nets, and Sketches

Mathematics

Class 5 | CBSE, NCERT & ICSE

Learning Objectives

Identify 3D shapes in everyday objects (cube, cuboid, cylinder, cone, sphere, prism, pyramid)
Understand the difference between 2D shapes and 3D shapes (solids)
Count faces, edges, and vertices of 3D shapes
Understand and draw the net of a cube, cuboid, cylinder, and cone
Identify which nets can fold into a given 3D shape
Draw different views of a 3D object (front, side, top)
Understand Euler's formula: F + V − E = 2
Build 3D shapes from their nets using paper

Introduction — 2D vs 3D

In earlier classes, you learned about 2D (flat) shapes like squares, rectangles, triangles, and circles. These have only length and width — they lie flat on paper. But the real world is 3D! A box, a ball, a tin can, an ice cream cone — these are all 3D shapes (also called solids). They have length, width, AND height (or depth).

A net is what a 3D shape looks like when you "unfold" it flat. Imagine cutting open a cardboard box along its edges and flattening it — that flat pattern is the net of the box. Knowing about nets helps in packaging, gift wrapping, building, and even in space engineering (ISRO folds solar panels flat for launch, then unfolds them in space)!

Image needed: Comparison of 2D shapes vs 3D shapes: show a square next to a cube, a circle next to a sphere, a rectangle next to a cuboid, a triangle next to a triangular prism. Clear arrows labelling "2D (flat)" and "3D (solid)".
Style: Clean side-by-side comparison diagram, colourful
Size: 700×350px

Common 3D Shapes

Shape	Faces	Edges	Vertices	Real-Life Examples (India)
Cube	6 (all squares)	12	8	Rubik's cube, dice, sugar cube, Maggi bouillon cube
Cuboid	6 (rectangles)	12	8	Brick, book, tiffin box, shoe box, phone
Cylinder	3 (2 circles + 1 curved)	2	0	Water bottle, Bournvita tin, pillar, pipe, rolling pin
Cone	2 (1 circle + 1 curved)	1	1 (apex)	Ice cream cone, party hat, traffic cone, temple gopuram
Sphere	1 (all curved)	0	0	Cricket ball, globe, ladoo, marble, Earth
Triangular Prism	5 (2 triangles + 3 rectangles)	9	6	Toblerone box, tent, prism in science lab
Square Pyramid	5 (1 square + 4 triangles)	8	5	Pyramids of Egypt, roof of a hut

Image needed: All 7 common 3D shapes drawn clearly with labels: cube, cuboid, cylinder, cone, sphere, triangular prism, square pyramid. Each shape shown in 3D perspective with faces slightly shaded, edges visible, vertices marked with dots.
Style: Clean 3D diagrams with labels, multicolour shading
Size: 750×400px

Key Terms

Face: A flat surface of a 3D shape (can also be curved)
Edge: A line where two faces meet
Vertex (plural: vertices): A point where edges meet (a corner)

Euler's Formula

The Swiss mathematician Leonhard Euler discovered a beautiful relationship between faces, vertices, and edges of any polyhedron (a 3D shape with flat faces):

Euler's Formula: F + V − E = 2

Faces + Vertices − Edges = 2 (always!)

Let's verify for the shapes we know:

Shape	F (Faces)	V (Vertices)	E (Edges)	F + V − E
Cube	6	8	12	6 + 8 − 12 = 2 ✓
Cuboid	6	8	12	6 + 8 − 12 = 2 ✓
Triangular Prism	5	6	9	5 + 6 − 9 = 2 ✓
Square Pyramid	5	5	8	5 + 5 − 8 = 2 ✓
Triangular Pyramid (Tetrahedron)	4	4	6	4 + 4 − 6 = 2 ✓

Note: Euler's formula works for all polyhedra (flat-faced 3D shapes). It does NOT work for shapes with curved surfaces (sphere, cylinder, cone) — those are not polyhedra.

Nets of 3D Shapes

A net is a flat 2D pattern that can be folded to make a 3D shape. When you cut a 3D shape along some edges and unfold it, you get its net.

Net of a Cube

A cube has 6 square faces. Its net is 6 connected squares arranged so they can fold into a cube. There are exactly 11 different nets of a cube!

The most common net looks like a cross (+) with 4 squares in a row and 1 square on each side of the second square.

Image needed: Show the 11 different nets of a cube, each drawn as connected squares on a grid. Number them 1–11. Also show one net with an arrow indicating how it folds into a cube.
Style: Grid-based diagrams, each net in a different colour, compact arrangement
Size: 700×500px

Net of a Cuboid

A cuboid has 6 rectangular faces: 2 each of 3 different sizes (length×width, length×height, width×height). Its net shows all 6 rectangles connected.

Example 1: A cuboid is 5 cm long, 3 cm wide, and 2 cm tall. Describe its net.

Solution: The net will have:

2 rectangles of 5 cm × 3 cm (top and bottom)
2 rectangles of 5 cm × 2 cm (front and back)
2 rectangles of 3 cm × 2 cm (left and right sides)

These 6 rectangles are arranged connected along their edges so they fold into the cuboid.

Net of a Cylinder

A cylinder unfolds into: 2 circles (top and bottom) + 1 rectangle (the curved surface unfurled). The rectangle's width = height of cylinder, and its length = circumference of the circular base (2πr).

Net of a Cone

A cone unfolds into: 1 circle (the base) + 1 sector (fan-shaped piece from the curved surface).

Image needed: Four net diagrams side by side: (1) Cube net (cross shape), (2) Cuboid net, (3) Cylinder net (2 circles + rectangle), (4) Cone net (circle + sector). Each with dotted fold lines and an arrow showing the folded 3D shape.
Style: Clean diagrams with fold lines, each net in a different colour
Size: 750×400px

Views of 3D Shapes

When you look at a 3D object from different directions, you see different 2D shapes. The three standard views are:

View	Direction	What You See
Front view	Looking at the object from the front	Height × Width (face facing you)
Side view	Looking from the left or right side	Height × Depth
Top view (Plan)	Looking from directly above	Width × Depth

Example 2: What are the front, side, and top views of a cylinder standing upright?

Solution:

Front view: Rectangle (height × diameter)
Side view: Rectangle (same as front)
Top view: Circle

Example 3: What are the front, side, and top views of a cone standing on its base?

Solution:

Front view: Triangle
Side view: Triangle
Top view: Circle (with a dot in centre for the apex)

Image needed: A visual showing a cube, cylinder, and cone with arrows pointing to their front, side, and top views. Each 3D shape on the left, with 3 view boxes (Front, Side, Top) on the right showing the 2D shape you'd see.
Style: Three-column diagram for each shape, clear labels, educational
Size: 700×500px

Key Words

Word	Meaning
3D Shape / Solid	A shape with length, width, and height (takes up space)
Face	A flat (or curved) surface of a 3D shape
Edge	A line segment where two faces meet
Vertex	A point where three or more edges meet (corner)
Net	A 2D pattern that folds into a 3D shape
Polyhedron	A 3D shape with all flat faces (no curves)
Prism	A 3D shape with two identical parallel ends and rectangular sides
Pyramid	A 3D shape with a polygon base and triangular sides meeting at a point
Cross-section	The shape you get when you slice through a 3D object
Euler's Formula	F + V − E = 2 (for any polyhedron)

Key Points to Remember

3D shapes have length, width, AND height. 2D shapes have only length and width
A cube has 6 faces (squares), 12 edges, 8 vertices. A cuboid has 6 faces (rectangles)
A net is a flat pattern that folds into a 3D shape
A cube has 11 different possible nets
Euler's formula: F + V − E = 2 (works for all polyhedra)
Cylinder net = 2 circles + 1 rectangle. Cone net = 1 circle + 1 sector
Three standard views: Front, Side, Top (also called Plan)
A sphere has only 1 curved surface — no edges, no vertices
The cylinder's rectangle length = circumference = 2πr
Cross-sections help visualize what's inside a 3D shape

Finished learning? Now test what you know!

🎯 Quick Quiz — Test Yourself!

Answer these questions right here. You get instant feedback — no paper needed!

Score: 0 / 8

1. How many faces does a cube have?

A cube has 6 square faces.

2. How many edges does a cube have?

A cube has 12 edges.

3. How many vertices (corners) does a cube have?

A cube has 8 vertices.

4. A flat shape that can be folded to make a 3D solid is called a:

A net folds up into a 3D shape.

5. Which 3D shape looks like a tin can or a Bournvita jar?

A cylinder — two circular faces and one curved surface.

6. Which 3D shape is perfectly round like a cricket ball or a ladoo?

A sphere — it has one curved surface, no edges or vertices.

7. How many faces does a cuboid (like a brick) have?

A cuboid has 6 rectangular faces.

8. Which 3D shape looks like an ice-cream cone or a party hat?

A cone has one circular face and a curved surface meeting at a point (apex).

Practice Questions

A. Identify the 3D Shape

1. A cricket ball:

2. A brick:

3. An ice cream cone:

4. A Bournvita tin:

5. A Rubik's cube:

6. A Toblerone chocolate box:

7. The Great Pyramid of Egypt:

8. A tent:

B. Faces, Edges, and Vertices

1. Complete: A cube has faces, edges, vertices.

2. Complete: A cylinder has faces, edges, vertices.

3. Complete: A cone has faces, edges, vertices.

4. Complete: A sphere has faces, edges, vertices.

5. A triangular prism has 5 faces, 6 vertices, and how many edges? Verify with Euler's formula.

6. A hexagonal prism has 8 faces and 12 vertices. Use Euler's formula to find its edges.

C. Euler's Formula

1. A polyhedron has 8 faces and 6 vertices. Find its edges.

2. A polyhedron has 10 edges and 6 vertices. Find its faces.

3. A polyhedron has 7 faces and 10 vertices. Find its edges.

4. Is it possible to have a polyhedron with 4 faces, 4 vertices, and 8 edges? Why?

5. A football (truncated icosahedron) has 32 faces and 60 vertices. How many edges?

D. Nets — Will It Fold?

1. Draw 3 different nets that can fold into a cube.

2. A flat pattern has 5 rectangles and 2 triangles. What 3D shape will it make?

3. A flat pattern has 2 circles and 1 rectangle. What 3D shape will it make?

4. A flat pattern has 1 circle and 1 sector (fan shape). What 3D shape?

5. Can 4 squares arranged in a straight line (1×4) fold into a cube? Why or why not?

6. Draw the net of a cuboid that is 4 cm long, 3 cm wide, and 2 cm tall.

E. Views of 3D Shapes

1. Front, side, and top views of a cube (all the same — what shape?):

2. A sphere looks like a from every direction.

3. A cylinder standing upright — top view: , front view:

4. A cone standing on base — top view: , front view:

5. Two cubes stacked on top of each other — front view: , side view: , top view:

F. Multiple Choice

1. Which 3D shape has no flat faces at all?

a) Cubeb) Cylinderc) Sphered) Cone

2. How many edges does a cube have?

a) 6b) 8c) 12d) 10

3. A cube has how many different nets?

a) 6b) 8c) 11d) 14

4. The net of a cylinder includes:

a) 6 squaresb) 2 circles + 1 rectanglec) 4 triangles + 1 squared) 1 circle + 1 triangle

5. Using Euler's formula, if F=6 and V=8, then E=?

a) 10b) 12c) 14d) 16

G. Draw and Create

1. Draw the net of a cube with each face 3 cm × 3 cm. Cut it out and fold it.

2. Draw the net of a triangular prism (equilateral triangle base with 4 cm side, height 6 cm).

3. Draw the front, side, and top views of a book lying on a table.

4. Draw the front, side, and top views of 3 cubes placed in an L-shape.

5. Draw a 3D sketch of a cube, a cuboid, and a square pyramid (show depth using dotted lines for hidden edges).

Activity: Build 3D Shapes from Nets

Materials: Thick chart paper, scissors, ruler, glue, pencil, compass (for circles)

Instructions:

Draw the net of a cube (6 connected squares, each 5 cm × 5 cm) on thick paper. Add small tabs (1 cm flaps) on alternate edges for gluing.
Cut out the net carefully along the outer edges.
Score (lightly press a ruler and pencil along) each fold line.
Fold all edges inward along the score lines.
Apply glue to the tabs and press them inside to form the cube.
Repeat for a cuboid (4 cm × 3 cm × 2 cm) and a triangular prism.

Challenge: Decorate each face with numbers (like a die) or letters. Can you make a fair die where opposite faces always add up to 7?

Image needed: Step-by-step photos/illustrations of a child building a cube from its net: drawing the net on paper, cutting it, scoring fold lines, folding, and gluing. Final cube shown decorated as a die. Indian student in school uniform.
Style: Step-by-step instructional illustration, 5 panels
Size: 750×350px

Want to review the lesson again?

Mathematics

3D Shapes: Boxes, Nets, and Sketches

Class 5 | CBSE / NCERT / ICSE

Name: Class: Date:

3D shapes (solids) have length, width, and height. Unlike 2D shapes, they take up space. A net is a flat pattern that folds into a 3D shape. Understanding nets helps in packaging, architecture, and engineering (ISRO folds solar panels flat for launch!).

Common 3D Shapes — Faces, Edges, Vertices

Shape	Faces	Edges	Vertices	Examples
Cube	6 (squares)	12	8	Dice, Rubik's cube
Cuboid	6 (rectangles)	12	8	Brick, book, tiffin box
Cylinder	3 (2 circles + 1 curved)	2	0	Water bottle, pipe
Cone	2 (1 circle + 1 curved)	1	1	Ice cream cone, party hat
Sphere	1 (curved)	0	0	Ball, globe, ladoo
Triangular Prism	5	9	6	Toblerone box, tent
Square Pyramid	5	8	5	Pyramids of Egypt

All 7 common 3D shapes with labels: cube, cuboid, cylinder, cone, sphere, prism, pyramid

Euler's Formula

F + V − E = 2 (for all polyhedra — 3D shapes with flat faces)

Faces + Vertices − Edges = 2

Verify: Cube → 6 + 8 − 12 = 2 ✓ | Pyramid → 5 + 5 − 8 = 2 ✓ | Prism → 5 + 6 − 9 = 2 ✓

Nets of 3D Shapes

Shape	Net Components	How Many Different Nets?
Cube	6 connected squares	11 different nets
Cuboid	6 connected rectangles (3 pairs)	54 different nets
Cylinder	2 circles + 1 rectangle	—
Cone	1 circle + 1 sector	—

Nets of cube, cuboid, cylinder, and cone with fold arrows

Views of 3D Shapes

Shape	Front View	Side View	Top View
Cube	Square	Square	Square
Cylinder (standing)	Rectangle	Rectangle	Circle
Cone (standing)	Triangle	Triangle	Circle (dot at centre)
Sphere	Circle	Circle	Circle

Mathematics

3D Shapes — Practice

Class 5 | CBSE / NCERT / ICSE

A. Identify the Shape

1. Cricket ball:

2. Brick:

3. Ice cream cone:

4. Bournvita tin:

5. Rubik's cube:

6. Tent (triangular):

7. Great Pyramid:

8. Die (dice):

B. Fill in the Blanks (F, E, V)

1. Cube: faces, edges, vertices

2. Cylinder: faces, edges, vertices

3. Cone: faces, edges, vertices

4. Sphere: faces, edges, vertices

5. Square Pyramid: faces, edges, vertices

C. Euler's Formula

1. F = 8, V = 6. Edges =

2. E = 10, V = 6. Faces =

3. F = 7, E = 15. Vertices =

4. Football: F = 32, V = 60. Edges =

5. Can F=4, V=4, E=8 exist? Why?

D. Nets

1. 2 circles + 1 rectangle → folds into:

2. 1 circle + 1 sector → folds into:

3. 6 connected squares → folds into:

4. Can 4 squares in a straight line fold into a cube?

5. A cube has how many different possible nets?

E. Views

1. Cube: Top view = , Front =

2. Cylinder (standing): Top = , Front =

3. Cone (standing): Top = , Front =

4. Sphere: Every view =

F. Multiple Choice

1. No flat faces at all: a) Cubeb) Cylinderc) Sphered) Cone

2. Cube edges: a) 6b) 8c) 12d) 10

3. Different cube nets: a) 6b) 8c) 11d) 14

4. Cylinder net: a) 6 squaresb) 2 circles + rectanglec) triangles + squared) circle + triangle

5. F=6, V=8 → E: a) 10b) 12c) 14d) 16

Mathematics

3D Shapes — Drawing & Building

Class 5 | CBSE / NCERT / ICSE

G. Draw

1. Draw 3 different nets that fold into a cube (use the grid below):

2. Draw the net of a cuboid (4 cm × 3 cm × 2 cm):

3. Draw front, side, and top views of a book lying flat on a table:

Front

Side

Top

4. Draw a 3D sketch of a cube (show hidden edges with dotted lines):

Activity: Build 3D Shapes from Nets

Materials: Thick paper, scissors, ruler, glue. Draw cube net (6 squares, 5 cm each), add 1 cm tabs on alternate edges. Cut, score fold lines, fold inward, glue. Make a cuboid and triangular prism too. Challenge: Decorate as a die (opposite faces = 7)!

Step-by-step building a cube from its net: draw, cut, fold, glue

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