Visualising Solid Shapes: CBSE Class 8 Maths NCERT Guide

Understanding Solid Shapes: 2D vs. 3D

What's the difference between a photograph of a box and the box itself? The photograph is a two-dimensional (2D) shape. It only has length and width. You can draw it on a piece of paper. The actual box, however, is a three-dimensional (3D) shape. It has length, width, and height. You can hold it, and it takes up space. These 3D shapes are also called solid shapes.

Every solid shape has three key parts:

• Faces: These are the flat surfaces of the solid. Think of the six sides of a dice; each one is a face.
• Edges: These are the line segments where two faces meet. On a dice, the lines where the sides join are the edges.
• Vertices: These are the corners of the solid, where three or more edges meet. A dice has 8 corners, or vertices.

Shapes that have flat faces, straight edges, and sharp vertices are called polyhedrons. Examples include cubes, cuboids, prisms, and pyramids. Shapes with curved surfaces, like spheres, cones, and cylinders, are not polyhedrons.

Key Terms for Solid Shapes

Face

A flat surface of a polyhedron. For example, a cube has 6 square faces.

Edge

A line segment formed by the intersection of two faces. A cube has 12 edges.

Vertex (plural: Vertices)

A point where three or more edges meet. It's essentially a corner. A cube has 8 vertices.

Polyhedron

A 3D solid figure whose surfaces are all flat polygons. Spheres and cylinders are not polyhedrons because they have curved surfaces.

Prism

A polyhedron with two identical and parallel bases, and whose other faces (lateral faces) are parallelograms. Example: Triangular prism.

Pyramid

A polyhedron with a polygon base and triangular faces that meet at a single point called the apex. Example: Square pyramid.

How to Visualise Different Views of a 3D Object

1. Step 1: Understand the Three Main Views — Imagine you have an object, like a toy car. We can look at it from three main directions: Front View: What you see when you look at the car from the front. Top View: What you see when you look down on the car from directly above. * Side View: What you see when you look at the car from the left or right side.
2. Step 2: Isolate a View and Sketch It — Let's take a simple example: a water bottle standing upright. Front View: It will look like a rectangle with a smaller rectangle/trapezium on top for the cap. Top View: It will look like a circle (for the cap). * Side View: It will look identical to the front view, a rectangle with a shape for the cap.
3. Step 3: Combine Views to Understand the Object — Architects and engineers use these views (called orthographic projections) to represent complex objects on a 2D plane. By looking at the front, top, and side views together, you can build a complete mental picture of the 3D shape.

Applying Euler's Formula (F + V - E = 2)

• A Swiss mathematician named Leonhard Euler discovered a fascinating relationship for all polyhedrons. It states that the number of Faces (F) plus the number of Vertices (V) minus the number of Edges (E) always equals 2. Let's test it!

  Example 1: A Cuboid (like a duster or a brick)

  • Step 1: Count the Faces (F). A cuboid has a top and bottom, and four sides. So, F = 6.
  • Step 2: Count the Vertices (V). A cuboid has four corners on the top and four corners on the bottom. So, V = 8.
  • Step 3: Count the Edges (E). There are 4 edges on the top, 4 on the bottom, and 4 connecting them. So, E = 12.
  • Step 4: Apply Euler's Formula. F + V - E = 6 + 8 - 12 = 14 - 12 = 2.
  • Result: The formula holds true!

• Example 2: A Triangular Pyramid (a pyramid with a triangular base)

  • Step 1: Count the Faces (F). It has one triangular base and three triangular sides. So, F = 4.
  • Step 2: Count the Vertices (V). It has three vertices at the base and one at the top (apex). So, V = 4.
  • Step 3: Count the Edges (E). There are three edges forming the base and three edges connecting the base to the apex. So, E = 6.
  • Step 4: Apply Euler's Formula. F + V - E = 4 + 4 - 6 = 8 - 6 = 2.
  • Result: The formula works for this shape too!

Exam Tip: Count Systematically to Avoid Errors

A very common mistake in exams is miscounting the faces, vertices, or edges of a polyhedron, especially for complex shapes. To avoid this, use a system!

• For Edges: First, count all the edges of the base. Then, count all the edges of the top face (if it's a prism). Finally, count the vertical or slanted edges that connect the bases or connect the base to the apex. This prevents double-counting.
• For Vertices: Count the vertices on the bottom base, then the vertices on the top face or at the apex.

This methodical approach will make sure your counts are accurate when you apply Euler's formula.

Practice Questions with Solutions

• Q: A polyhedron has 7 faces and 10 vertices. How many edges does it have?
  A: Step 1: Recall Euler's formula for polyhedrons: F + V - E = 2, where F is faces, V is vertices, and E is edges.
  Step 2: Substitute the given values into the formula. We have F = 7 and V = 10. So, 7 + 10 - E = 2.
  Step 3: Simplify the equation: 17 - E = 2.
  Step 4: Solve for E. To find E, we can rearrange the equation: E = 17 - 2. Therefore, E = 15.
  Final answer: The polyhedron has 15 edges.

• Q: Verify Euler's formula for a square pyramid.
  A: Step 1: Identify the faces (F), vertices (V), and edges (E) of a square pyramid.
  Faces (F): It has 1 square base and 4 triangular side faces. So, F = 1 + 4 = 5.
  Vertices (V): It has 4 vertices at the corners of the square base and 1 vertex at the top (apex). So, V = 4 + 1 = 5.
  Edges (E): It has 4 edges forming the square base and 4 edges connecting the base corners to the apex. So, E = 4 + 4 = 8.
  Step 2: Substitute these values into Euler's formula (F + V - E = 2).
  5 + 5 - 8 = 10 - 8 = 2.
  Step 3: Since the result is 2, Euler's formula is verified for a square pyramid.
  Final answer: For a square pyramid, F=5, V=5, E=8. F + V - E = 5 + 5 - 8 = 2. The formula is verified.

• Q: Can a polyhedron have 10 faces, 20 edges, and 15 vertices?
  A: Step 1: We need to check if these values satisfy Euler's formula: F + V - E = 2.
  Step 2: Substitute the given values: F = 10, V = 15, E = 20.
  Step 3: Calculate the value of the expression F + V - E.
  10 + 15 - 20 = 25 - 20 = 5.
  Step 4: Compare the result with 2. Since the result is 5, and not 2, these values do not satisfy Euler's formula.
  Final answer: No, a polyhedron cannot have 10 faces, 20 edges, and 15 vertices because it violates Euler's formula.

• Q: Draw the top view, front view, and side view of a dice.
  A: Step 1: Imagine a standard dice placed on a table. The front view is what you see looking at one face directly. The top view is what you see looking down from above. The side view is what you see from the left or right.
  Step 2: Draw the Front View. A dice is a cube, so its front view is a square. If the front face has '3' dots, you would draw a square with three dots.
  Step 3: Draw the Top View. Looking down from above, you will also see a square. If the top face has '1' dot, you would draw a square with one dot in the center.
  Step 4: Draw the Side View. Looking from the side, you will again see a square. If the side face has '5' dots, you would draw a square with five dots.
  Final answer: All three views (top, front, and side) of a dice are squares. The number of dots on them will depend on the orientation of the dice.

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