2-D shapes and 3-D objects
A page of your book is flat, but your school bag is a solid thing you can hold. The page is like a 2-D shape and the bag is like a 3-D object.
A 2-D (flat) shape has only length and breadth, like a square or circle. A 3-D (solid) object has length, breadth and height, like a cube or a ball.
In the figure : Everyday solids: cube, cylinder, cone and sphere.
Rule 1: Match objects with solids — Look at the object and compare it with a basic solid.
- Easy : A dice is a cube. A brick is a cuboid. A ball is a sphere.
- Medium : A candle is a cylinder. A party hat is a cone.
- Exam-type : A tent with triangular ends is like a triangular prism.
Rule 2: 2-D or 3-D? — Ask: can I hold it and does it have height? If yes, it is 3-D.
- Easy : A circle drawn on paper is 2-D; a ball is 3-D.
- Medium : A shadow on a wall is flat (2-D), although the object is solid (3-D).
- Exam-type : A drawing of a cube on paper is a 2-D picture of a 3-D object.
A solid has length, breadth and height. A flat shape has no height.
Saying a drawing of a cube is a solid. The paper drawing is flat; it only shows the cube.
Memory clue: 2 dimensions = flat; 3 dimensions = solid.
Spot the difference
| Name | Dimensions |
|---|---|
| Square, circle | 2-D |
| Cube, cuboid | 3-D |
| Ball | 3-D |
Practice questions
- 1What solid is a dice?
- 2What solid is a brick?
- 3What solid is a football?
- 4What solid is a party hat?
- 5Is a circle 2-D or 3-D?
- 6Is a candle 2-D or 3-D? Name its shape.
- 7What is the shape and dimension of a ball's shadow on a wall?
- 8Raju says a picture of a cube on paper is a 3-D object. Correct him.
Show answers (try them yourself first)
- 1Cube
- 2Cuboid
- 3Sphere
- 4Cone
- 52-D
- 63-D; cylinder
- 7A circle, which is 2-D.
- 8The paper picture is flat (2-D). It only shows a 3-D cube.
Faces, edges and vertices
A matchbox has flat sides, long edges where two sides meet and sharp corners. Every box has these three parts.
A flat surface is a face (पृष्ठ). The line where two faces meet is an edge (कडा). The corner point where edges meet is a vertex (शिरोबिंदू). A cube and a cuboid have 6 faces, 12 edges and 8 vertices.
In the figure : A cube has 6 faces, 12 edges and 8 vertices.
Rule 1: Cube and cuboid faces — A cube has 6 equal square faces. A cuboid has rectangular faces; opposite faces are equal.
- Easy : A cuboid 6 cm × 4 cm × 3 cm has 6 faces: two of 6 × 4, two of 6 × 3 and two of 4 × 3.
- Medium : A cuboid 4 cm × 4 cm × 7 cm has two square faces (4 × 4) and four rectangles (4 × 7).
- Exam-type : A cube has 3 edges meeting at every vertex; 8 vertices × 3 ÷ 2 = 12 edges.
Rule 2: Total length of edges — Cube of edge a: 12 × a. Cuboid l, b, h: 4 × (l + b + h).
- Easy : Cube of edge 6 cm: 12 × 6 = 72 cm.
- Medium : Cuboid 5, 4, 2 cm: 4 × (5 + 4 + 2) = 4 × 11 = 44 cm.
- Exam-type : A wire of 96 cm bent to make a cube frame: edge = 96 ÷ 12 = 8 cm.
Faces, edges and vertices
In the app you can move this figure with your finger.
Open in the appA cube and a cuboid both have 6 faces, 12 edges and 8 vertices.
Counting only the edges you can see in a picture. A cube has 12, including 3 hidden ones.
Memory clue: 6 – 12 – 8 : Faces, Edges, Vertices (F-E-V in order).
Spot the difference
| Part | Cube / cuboid |
|---|---|
| Faces | 6 |
| Edges | 12 |
| Vertices | 8 |
Practice questions
- 1How many faces does a cuboid have?
- 2How many edges does a cube have?
- 3How many vertices does a cuboid have?
- 4Total edge length of a cube of edge 6 cm.
- 5Total edge length of a cuboid 5 cm × 4 cm × 2 cm.
- 6A cuboid 4 × 4 × 7 cm: how many faces are squares and how many rectangles?
- 7A wire 96 cm long makes a cube frame. Find the edge.
- 8Sita says a cube has 8 edges since it has 8 vertices. What is wrong?
Show answers (try them yourself first)
- 16
- 212
- 38
- 412 × 6 = 72 cm
- 54 × 11 = 44 cm
- 62 squares and 4 rectangles
- 796 ÷ 12 = 8 cm
- 8Edges and vertices are different. A cube has 12 edges and 8 vertices.
Cylinder, cone, sphere, prism and pyramid
A funnel, a tin can and a ball all curve, while a tent and a gift box have flat sides. Count the faces, edges and vertices to know each solid.
A cylinder has 2 flat circular faces, 1 curved surface and no vertex. A cone has 1 flat face, 1 curved surface and 1 vertex. A sphere has only a curved surface.
In the figure : Faces, edges and vertices of common solids.
Rule 1: Solids with flat faces — Square pyramid 5 F, 8 E, 5 V. Triangular prism 5 F, 9 E, 6 V. Triangular pyramid 4 F, 6 E, 4 V.
- Easy : Triangular pyramid: 4 triangle faces, 6 edges, 4 vertices.
- Medium : Pentagonal prism: 7 faces, 15 edges, 10 vertices.
- Exam-type : Hexagonal pyramid: 7 faces, 12 edges, 7 vertices.
Rule 2: Euler's rule — For a solid with flat faces, vertices − edges + faces = 2.
- Easy : Cube: 8 − 12 + 6 = 2.
- Medium : Triangular prism: 6 − 9 + 5 = 2.
- Exam-type : A solid has 6 vertices and 9 edges. Faces = 2 − 6 + 9 = 5.
Curved solids (cylinder, cone, sphere) do not follow Euler's rule.
Counting the apex of a cone as an edge. A cone has 1 vertex and no straight edge.
Memory clue: V − E + F = 2 (Vertices, Edges, Faces in this order).
Spot the difference
| Solid | F / E / V |
|---|---|
| Triangular pyramid | 4 / 6 / 4 |
| Square pyramid | 5 / 8 / 5 |
| Triangular prism | 5 / 9 / 6 |
Practice questions
- 1How many vertices does a cone have?
- 2How many flat faces does a cylinder have?
- 3How many faces does a sphere have?
- 4Give the F, E, V of a triangular pyramid.
- 5Check Euler's rule for a square pyramid.
- 6A solid has 6 vertices and 9 edges. Find the faces and name it.
- 7Find the faces, edges and vertices of a hexagonal pyramid and check Euler's rule.
- 8Meena says a cone has 2 vertices. What is the right number?
Show answers (try them yourself first)
- 11
- 22
- 3None; it has only 1 curved surface.
- 44, 6, 4
- 5V − E + F = 5 − 8 + 5 = 2
- 6F = 2 − 6 + 9 = 5; triangular prism
- 77 faces, 12 edges, 7 vertices; 7 − 12 + 7 = 2.
- 81 vertex, at the tip. The base is a circle, not a corner.
Nets of solids
Open an empty cereal box carefully and lay it flat. The flat pattern you get is a net. Fold it back and you again get the box.
A net (जाळे) is a flat pattern that folds to make a solid. A cube's net has 6 squares; a cuboid's net has 6 rectangles.
In the figure : Six squares in a cross pattern fold to make a cube.
Rule 1: Nets of cube and cuboid — Cube: 6 equal squares. Cuboid: 6 rectangles in 3 equal pairs.
- Easy : A cube net has 6 squares.
- Medium : A cuboid net has 6 rectangles. Opposite faces match in size.
- Exam-type : A row of 6 squares cannot fold into a cube, because the squares would overlap and not close the box.
In the figure : The net of a square pyramid is one square and four triangles.
Rule 2: Nets of other solids — Cylinder: 1 rectangle + 2 circles. Square pyramid: 1 square + 4 triangles. Triangular prism: 2 triangles + 3 rectangles.
- Easy : A tin can: the label is the rectangle, top and bottom are the 2 circles.
- Medium : A net with 2 triangles and 3 rectangles folds to a triangular prism.
- Exam-type : Surface area of a cube of edge 3 cm: 6 × 3 × 3 = 54 cm².
Nets of solids
In the app you can move this figure with your finger.
Open in the appTotal area of a net = sum of all its faces. Cuboid 5 × 3 × 2 cm: 2 × (15 + 6 + 10) = 62 cm².
Counting fewer faces in the net. A cube always needs exactly 6 squares.
Memory clue: Net = unfolded box: Flatten – Fold – Form.
Spot the difference
| Solid | Net |
|---|---|
| Cube | 6 squares |
| Cylinder | 1 rectangle + 2 circles |
| Square pyramid | 1 square + 4 triangles |
Practice questions
- 1How many squares are in the net of a cube?
- 2How many rectangles are in the net of a cuboid?
- 3Which shapes make the net of a cylinder?
- 4A net has 1 square and 4 triangles. Which solid is it?
- 5A net has 2 triangles and 3 rectangles. Which solid is it?
- 6Find the surface area of a cube of edge 3 cm.
- 7Find the surface area of a cuboid 5 cm × 3 cm × 2 cm.
- 8Can a row of 6 squares fold into a cube? Give a reason.
Show answers (try them yourself first)
- 16
- 26
- 31 rectangle and 2 circles
- 4Square pyramid
- 5Triangular prism
- 66 × 9 = 54 cm²
- 72 × (15 + 6 + 10) = 62 cm²
- 8No. The squares overlap and the box does not close.
Views and solid problems
Put a tin of paint on a table and look from above and from the side. You see a circle from above and a rectangle from the side. Different views of one solid look different.
A solid looks different from the top, the side and the front. A box filled with unit cubes: number of cubes = length × breadth × height.
Rule 1: Views of solids — Cube: square from every side. Cylinder standing: circle from the top, rectangle from the side. Cone: circle from the top, triangle from the side. Sphere: circle from everywhere.
- Easy : A dice looks like a square from the top.
- Medium : A candle standing up looks like a circle from the top and a rectangle from the side.
- Exam-type : A party hat looks like a triangle from the side and a circle from the top.
Rule 2: Counting cubes — Cubes = l × b × h, or layer × number of layers.
- Easy : A box 5 × 4 × 3 cm holds 5 × 4 × 3 = 60 cubes of 1 cm.
- Medium : One layer has 4 × 3 = 12 cubes; 5 layers = 60 cubes.
- Exam-type : 100 cubes exactly fill a 5 × 5 × 4 box (5 × 5 × 4 = 100).
Identify a solid from its numbers: 5 F, 8 E, 5 V is a square pyramid; 2 flat faces, 1 curved surface, no vertex is a cylinder.
Adding l + b + h to count cubes. It must be multiplied.
Memory clue: Top view = base; side view = height.
Spot the difference
| Solid | Top / Side view |
|---|---|
| Cube | square / square |
| Cylinder | circle / rectangle |
| Cone | circle / triangle |
Practice questions
- 1What does a dice look like from the top?
- 2What does a cone look like from the side?
- 3What does a ball look like from every side?
- 4How many 1 cm cubes fill a box 5 × 4 × 3 cm?
- 5A solid has 2 flat faces, 1 curved surface and no vertex. Name it.
- 6A solid has 5 faces, 8 edges and 5 vertices. Name it.
- 7The bottom layer of a box has 4 × 3 cubes and there are 5 layers. How many cubes?
- 8Meena has 100 cubes of 1 cm. Can she fill a box 5 × 5 × 4 cm exactly? Show how.
Show answers (try them yourself first)
- 1A square
- 2A triangle
- 3A circle
- 460
- 5Cylinder
- 6Square pyramid
- 712 × 5 = 60
- 8Yes, 5 × 5 × 4 = 100 cubes.