Three-dimensional Shapes Chapter Notes | Mathematics Class 6 ICSE PDF Download

Introduction

Imagine the world around you—everything from a tiny thread to a massive wardrobe has a shape! Some objects, like a single strand of hair, only stretch in one direction (length). Others, like a piece of paper, spread out in two directions (length and breadth). But the most fascinating ones, like a box or a ball, take up space in three directions—length, breadth, and height. These are called three-dimensional (3D) shapes, and they’re all around us in our daily lives, from the juice can you drink from to the dice you roll in a game. In this chapter, we’ll dive into the exciting world of 3D shapes, exploring their parts, types, and how they can be unfolded or viewed from different angles. Get ready to see the world in a whole new dimension!

Classification of Objects: 1D, 2D, and 3D

• Objects are classified based on the dimensions they occupy.
• One-dimensional (1D): Objects with only length, like a thread or a hair.
• Two-dimensional (2D): Objects with length and breadth, like a sheet of paper.
• Three-dimensional (3D): Objects with length, breadth, and height, like a wardrobe or a box.
• 3D shapes are the focus of this chapter, as they have volume and take up space in all three directions.

Components of a 3D Shape

• 3D shapes have three main parts: faces, edges, and vertices.
• Face (F): A flat or curved surface of a 3D shape.
• Edge (E): The line where two faces meet.
• Vertex (V): The point where two or more edges intersect, also called a corner.

Polyhedrons

• A polyhedron is a 3D shape made up of flat polygonal faces.
• Polygons are 2D shapes with straight sides, like triangles or squares.
• Polyhedrons have only flat faces, no curved surfaces.

Cuboid

• A cuboid is a 3D shape with six rectangular faces.
• All faces are rectangles, and opposite faces are equal in size.
• It has 6 faces, 12 edges, and 8 vertices.
  

Cube

• A cube is a special cuboid where all six faces are identical squares.
• It has 6 faces, 12 edges, and 8 vertices.
• All edges are of equal length.
  

Pyramid

• A pyramid is a polyhedron with a polygonal base and triangular side faces that meet at a common point called the apex.
• The base can be any polygon, like a square or triangle.
• The side faces are always triangles.

Square Pyramid

• A pyramid with a square base is called a square pyramid.
• It has 5 faces (1 square base and 4 triangular side faces), 8 edges, and 5 vertices.

Triangular Pyramid or Tetrahedron

• A pyramid with a triangular base is called a triangular pyramid or tetrahedron.
• It has 4 faces (all triangles), 6 edges, and 4 vertices.
• All faces are triangular and equal in a regular tetrahedron.

Prism

• A prism is a polyhedron with two parallel, congruent polygonal bases connected by parallelogram side faces.
• The bases can be any polygon, like triangles or squares.
• The side faces are parallelograms.

Triangular Prism

• A prism with triangular bases is called a triangular prism.
• It has 5 faces (2 triangular bases and 3 rectangular side faces), 9 edges, and 6 vertices.

3D Shapes with Curved Face

• Some 3D shapes have curved surfaces instead of flat polygonal faces.
• These include spheres, cones, and cylinders.

Sphere

• A sphere is a 3D shape where every point on the surface is equidistant from its center.
• It has 1 curved face, 0 edges, and 0 vertices.
• Formula:
  • Surface Area = 4πr² (where r is the radius).
  • Volume = (4/3)πr³.

Cone

• A cone is a 3D shape with a single circular base that tapers to a point called the apex.
• It has 1 curved face, 1 flat circular base, 1 edge, and 1 vertex.
• Formula:
  • Curved Surface Area = πrl (where r is the radius of the base, l is the slant height).
  • Total Surface Area = πr(r + l).
  • Volume = (1/3)πr²h (where h is the height).

Cylinder

• A cylinder is a 3D shape with two parallel circular bases connected by a curved surface.
• It has 1 curved face, 2 flat circular faces, and 2 edges.
• Formula:
  • Curved Surface Area = 2πrh (where r is the radius, h is the height).
  • Total Surface Area = 2πr(r + h).
  • Volume = πr²h.

Nets of 3D Shapes

• A net is a 2D shape that can be folded to form a 3D shape.
• It is like the unfolded "skeleton" of a 3D object.
• Not all 3D shapes have nets (e.g., a sphere does not have a net).

Nets of Various 3D Shapes

Cube

• A cube’s net consists of 6 squares arranged in a specific pattern.
• Common arrangements include a cross pattern or a 2x3 grid of squares.

Cuboid

• A cuboid’s net consists of 6 rectangles arranged to form the shape when folded.
• The rectangles are arranged to match the cuboid’s dimensions (length, breadth, height).

Cylinder

• A cylinder’s net includes 2 circles (for the bases) and a curved rectangle (for the curved surface).
• The rectangle’s length equals the circumference of the base (2πr).

Cone

• A cone’s net consists of 1 circle (for the base) and a sector of a larger circle (for the curved surface).
• The sector’s arc length equals the circumference of the base (2πr).

Triangular Pyramid or Tetrahedron

• A tetrahedron’s net consists of 4 triangles arranged so that they fold to form a pyramid with a triangular base.
• The triangles must be arranged to share edges correctly.

Visualizing Solid Objects

• Visualizing a 3D shape means imagining how it looks from different angles: front, side, or top.
• Not all parts of a 3D shape are visible from one angle, especially in combined shapes where some parts may be hidden.
• Different views (front, side, top) show different aspects of the shape.

Solved Examples