K2KARAKORAM
मराठीEnglish
Open the app

State Board · Class 6 · Mathematics

Lesson 1: Three-Dimensional Shapes

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Dimensions of shapes

From daily life

A line drawn on paper has only length. The paper itself has length and width. A brick has length, width and also height, so we can hold it. That extra measure makes it a solid shape.

Simple definition

A shape's dimension (मिती) tells how many measures it has: 1D has length, 2D has length and width, 3D has length, width and height.

Rule 1: 1D, 2D and 3D — A line segment is 1D. A square or a rectangle is 2D; it lies flat on a table. A cuboid (इष्टिकाचिती), cylinder (दंडगोल), cone (शंकू) and sphere (गोल) are 3D because they take up space.

  • Easy : Easy: A pencil line is 1D, a page is 2D, a dice is 3D.
  • Medium : Medium: A square drawn on a notebook is 2D, not 3D.
  • Exam-type : Exam: Which has length, width and height: a rectangle or a box? The box.

Rule 2: Everyday 3D objects — A matchbox is a cuboid. A stack of carrom pieces is a cylinder. A funnel or an ice-cream cone is a cone. A ball or a ladoo is a sphere.

CubeCylinderConeSphere

In the figure : In the figure four common 3D shapes: a cuboid, a cylinder, a cone and a sphere.

  • Easy : Easy: A matchbox is a cuboid.
  • Medium : Medium: A ball is a sphere; a stack of coins is a cylinder.
  • Exam-type : Exam: A party hat is a cone; a square is not a 3D shape.
Remember

1D: length. 2D: length and width. 3D: length, width and height.

Common mistake

Calling a drawn square a 3D shape. A square is flat, so it is 2D.

Shortcut

Memory clue: "हातात धरता येते = 3D".

Spot the difference

DimensionMeasuresExample
1Dlengthline segment
2Dlength, widthrectangle
3Dlength, width, heightcuboid

Practice questions

Easy
  1. 1How many measures does a 2D shape have?
  2. 2Is a line segment 1D, 2D or 3D?
  3. 3Is a dice 3D?
Medium
  1. 4Name the shape of a matchbox.
  2. 5Name the shape of a ball.
  3. 6Name the shape of a stack of carrom pieces.
Challenging
  1. 7Is a square drawn in a notebook 2D or 3D?
  2. 8Why can we hold a 3D shape in the hand but not a 2D drawing?
Show answers (try them yourself first)
  1. 1Two: length and width.
  2. 21D.
  3. 3Yes, it has length, width and height.
  4. 4Cuboid.
  5. 5Sphere.
  6. 6Cylinder.
  7. 72D, because it is flat.
  8. 8A 3D shape has height and takes up space; a 2D drawing is flat.

Elements of shapes and polygons

From daily life

Look at a dice. The flat sides are its faces, the lines where two faces meet are edges, and the corners are vertices.

Simple definition

In 3D shapes: face (पृष्ठ) is a flat or curved surface, edge (कडा) is where two faces meet, vertex (शिरोबिंदू) is where edges meet.

Rule 1: 2D shapes and polygons — In a 2D shape, the common point of two sides is a vertex and a segment joining two adjacent vertices is a side. A polygon (बहुभुज) is a closed figure with three or more sides, and its vertices equal its sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7.

  • Easy : Easy: A triangle has 3 sides and 3 vertices.
  • Medium : Medium: A hexagon has 6 sides and 6 vertices.
  • Exam-type : Exam: A figure with 7 vertices has 7 sides: a heptagon.

Rule 2: Parts of a cuboid — A cuboid has 6 faces, 12 edges and 8 vertices. A 3D shape is made of many faces, and a vertex is where three or more edges meet.

faceedgevertex

In the figure : In the figure one face, one edge and one vertex of a cuboid are marked in gold.

Live figure

Elements of shapes and polygons

In the app you can move this figure with your finger.

Open in the app
  • Easy : Easy: A dice has 6 faces.
  • Medium : Medium: A dice has 12 edges and 8 vertices.
  • Exam-type : Exam: A cuboid has 6 + 8 = 14 faces and vertices together, and 12 edges.
Remember

Cuboid: 6 faces, 12 edges, 8 vertices. Polygon vertices = sides.

Common mistake

Mixing up edge and vertex. Edge is a line, vertex is a point.

Shortcut

Memory clue: "पृष्ठ सपाट, कडा रेघ, शिरोबिंदू टोक".

Spot the difference

PartMeaningCuboid
Faceflat surface6
Edgetwo faces meet12
Vertexedges meet8

Practice questions

Easy
  1. 1How many faces does a cuboid have?
  2. 2How many edges does a cuboid have?
  3. 3How many vertices does a cuboid have?
Medium
  1. 4What is a polygon?
  2. 5How many sides does a pentagon have?
  3. 6A polygon has 7 vertices. How many sides does it have?
Challenging
  1. 7What is an edge in a 3D shape?
  2. 8How many faces and vertices does a cuboid have together?
Show answers (try them yourself first)
  1. 16.
  2. 212.
  3. 38.
  4. 4A closed figure with three or more sides.
  5. 55.
  6. 67.
  7. 7The line where two faces meet.
  8. 86 + 8 = 14.

Prisms

From daily life

A tin of biscuits, a log of wood and a stack of coins are all made by placing equal shapes one on top of another.

Simple definition

A prism (प्रिझम) is formed by placing equal shapes on top of each other. Its vertical faces are rectangles.

Rule 1: Triangular and rectangular prisms — Triangles stacked give a triangular prism: 6 vertices, 9 edges, 2 triangular faces and 3 rectangular faces. Rectangles stacked give a cuboid (rectangular prism): 8 vertices, 12 edges and 6 rectangular faces.

  • Easy : Easy: A tent shaped like a roof ridge is a triangular prism.
  • Medium : Medium: A triangular prism has 2 + 3 = 5 faces.
  • Exam-type : Exam: A cuboid is a rectangular prism with 12 edges.

Rule 2: Cylinder and bigger prisms — Equal circles stacked give a cylinder (circular prism): 2 flat circular faces, 1 curved face and 2 circular edges. A pentagonal prism has 10 vertices, 15 edges (3 × 5) and 7 faces.

  • Easy : Easy: A cylinder has 2 flat faces and 1 curved face.
  • Medium : Medium: A hexagonal prism has 3 × 6 = 18 edges.
  • Exam-type : Exam: A pentagonal prism has 2 pentagons and 5 rectangles, so 7 faces.
Remember

Prism: two equal ends joined by rectangles. Edges = 3 × sides of the base.

Common mistake

Counting only the rectangles as faces. The two ends are faces too.

Shortcut

Memory clue: "प्रिझम : दोन टोके + बाजू आयत".

Spot the difference

PrismVerticesEdgesFaces
Triangular695
Cuboid8126
Pentagonal10157

Practice questions

Easy
  1. 1What is a prism?
  2. 2How many vertices does a triangular prism have?
  3. 3How many edges does a triangular prism have?
Medium
  1. 4How many faces does a triangular prism have?
  2. 5Describe the faces of a cylinder.
  3. 6How many edges does a hexagonal prism have?
Challenging
  1. 7How many vertices and faces does a pentagonal prism have?
  2. 8A prism has a base with 8 sides. How many edges does it have?
Show answers (try them yourself first)
  1. 1A solid made by placing equal shapes on top of each other.
  2. 26.
  3. 39.
  4. 45 (2 triangles and 3 rectangles).
  5. 52 flat circular faces and 1 curved face.
  6. 63 × 6 = 18.
  7. 710 vertices and 7 faces.
  8. 83 × 8 = 24.

Pyramids, cone and sphere

From daily life

A tent, the top of a pyramid in Egypt and a birthday cap all narrow to a point at the top. Some such shapes have a flat base with straight sides, some have a round base.

Simple definition

A pyramid (पिरॅमिड) has triangular faces meeting at one top vertex. A cone (शंकू) has a round base and one vertex. A sphere (गोल) has no edges or vertices.

Rule 1: Pyramids — A triangular pyramid has 4 vertices, 4 triangular faces and 6 edges. A square pyramid has 5 vertices, 4 triangular faces, 1 square face and 8 edges.

  • Easy : Easy: A square pyramid has 5 faces in all.
  • Medium : Medium: A pentagonal pyramid has 6 vertices and 2 × 5 = 10 edges.
  • Exam-type : Exam: A pyramid with a hexagonal base has 6 triangles + 1 hexagon = 7 faces and 12 edges.

Rule 2: Cone and sphere — A cone has 1 vertex, 1 curved surface, 1 circular edge and 1 flat circular face. Rolling a semicircle of radius 6 cm gives a cone with a base circle of radius 3 cm. A ball or a ladoo is a sphere: it has a curved surface only.

  • Easy : Easy: An ice-cream cone is a cone; a ladoo is a sphere.
  • Medium : Medium: A cone has 1 flat face and 1 curved face.
  • Exam-type : Exam: An hourglass is two similar cones joined at their vertices.
Remember

Pyramid edges = 2 × sides of the base. Sphere has no edge and no vertex.

Common mistake

Giving a sphere edges and vertices. A sphere is fully curved.

Shortcut

Memory clue: "पिरॅमिड : टोक एक, बाजू त्रिकोण".

Spot the difference

ShapeVerticesEdges
Triangular pyramid46
Square pyramid58
Cone11 circular
Sphere00

Practice questions

Easy
  1. 1How many vertices does a triangular pyramid have?
  2. 2How many edges does a triangular pyramid have?
  3. 3How many faces does a square pyramid have?
Medium
  1. 4How many edges does a square pyramid have?
  2. 5Does a sphere have any edge or vertex?
  3. 6Name the parts of a cone.
Challenging
  1. 7How many vertices and edges does a pentagonal pyramid have?
  2. 8A pyramid has a base with 9 sides. How many edges does it have?
Show answers (try them yourself first)
  1. 14.
  2. 26.
  3. 35 (4 triangles and 1 square).
  4. 48.
  5. 5No, it has only a curved surface.
  6. 61 vertex, 1 curved surface, 1 circular edge and 1 flat circular face.
  7. 76 vertices and 10 edges.
  8. 82 × 9 = 18.

Difference between prism and pyramid

From daily life

A Toblerone-shaped box and a tent both have a triangle, yet one has two triangle ends and the other has one pointed top. Counting edges tells them apart.

Simple definition

Prism: two equal ends, rectangular side faces, edges = 3 × sides of the base. Pyramid: one base, triangular side faces, edges = 2 × sides of the base.

Rule 1: Comparing — Triangular prism: 6 vertices, 9 edges, 3 rectangular faces, 2 triangular ends. Triangular pyramid: 4 vertices, 6 edges, 3 triangular side faces, 1 triangular base.

Prism: 12 edgesPyramid: 8 edges

In the figure : In the figure a prism (two equal ends joined by rectangles) next to a pyramid (one base and triangles meeting at the top).

  • Easy : Easy: Prism side faces are rectangles; pyramid side faces are triangles.
  • Medium : Medium: Square prism has 12 edges, square pyramid 8 edges.
  • Exam-type : Exam: A pyramid has 1 vertex at the top and the rest at the base.

Rule 2: Counting rules — If the base has n sides: a prism has 2n vertices, 3n edges and n + 2 faces; a pyramid has n + 1 vertices, 2n edges and n + 1 faces. For a hexagonal prism: 12 vertices, 18 edges, 8 faces. For a hexagonal pyramid: 7 vertices, 12 edges, 7 faces.

  • Easy : Easy: Pentagonal prism edges: 3 × 5 = 15.
  • Medium : Medium: Pentagonal pyramid edges: 2 × 5 = 10.
  • Exam-type : Exam: A solid has 18 edges and rectangular side faces. It is a prism with 18 ÷ 3 = 6 sides in the base.
Remember

Prism edges 3n, pyramid edges 2n.

Common mistake

Swapping 3n and 2n. Prism has two ends, so more edges.

Shortcut

Memory clue: "प्रिझम 3 ने, पिरॅमिड 2 ने".

Spot the difference

FeaturePrismPyramid
Base2 equal ends1
Side facesRectanglesTriangles
Edges3 × sides2 × sides

Practice questions

Easy
  1. 1Which has rectangular side faces: prism or pyramid?
  2. 2Which has triangular side faces?
  3. 3How many edges does a square prism have?
Medium
  1. 4How many edges does a square pyramid have?
  2. 5How many edges does a pentagonal prism have?
  3. 6How many vertices and faces does a hexagonal prism have?
Challenging
  1. 7How many vertices and faces does a hexagonal pyramid have?
  2. 8A prism has 18 edges. How many sides does its base have?
Show answers (try them yourself first)
  1. 1Prism.
  2. 2Pyramid.
  3. 33 × 4 = 12.
  4. 42 × 4 = 8.
  5. 515.
  6. 612 vertices and 8 faces.
  7. 77 vertices and 7 faces.
  8. 818 ÷ 3 = 6.

Quick revision

  • 1D: length; 2D: length and width; 3D: length, width and height.
  • Prism edges = 3 × sides of the base; pyramid edges = 2 × sides of the base.
  • Sphere: no edges and no vertices; cone: one vertex, one curved surface, one circular edge, one circular face.