Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
Three-Dimensional (3D) Shapes: Unlike 2D shapes which have only length and breadth, 3D shapes possess three dimensions: length, breadth, and height (or depth). Common examples include cubes, cuboids, spheres, cylinders, and cones.
Perspective and Multiple Views: A 3D object appears different depending on the position of the observer. To represent a 3D object on a 2D surface like paper, we draw its views from specific directions: the Top view, the Front view, and the Side view.
The Front View: This is the 2D shape seen when looking at an object directly from the front. For example, the front view of a vertical cylinder is a rectangle, and the front view of a house might show the door and front windows.
The Side View: This represents the object as seen from either the left or the right side. In a standard cuboid, the side view is a rectangle whose dimensions are the breadth and height of the cuboid. For a cone, the side view appears as a triangle.
The Top View (Plan): This is the view from directly above the object. For instance, the top view of a water bottle is usually a circle, while the top view of a table is typically a rectangle. This view is often used in maps and architectural plans.
Visualising Composite Solids: Objects made by joining multiple cubes can be visualised by looking at the arrangement of squares in each direction. If three cubes are stacked in a row, the front view is a rectangle of squares, while the side view is a single square.
Mapping Space: Maps differ from pictures as they show the relative location of objects rather than how they look to the eye. Maps use fixed scales and symbols to represent 3D features on a 2D plane, often correlating with a top-down perspective.
Polyhedrons and Euler's Formula: 3D solids with flat faces, straight edges, and vertices are called polyhedrons. The relationship between the number of faces (), vertices (), and edges () is given by the Euler’s formula: .
📐Formulae
Euler's Formula for any Polyhedron:
Where = Number of faces
Where = Number of vertices
Where = Number of edges
Scale of a map =
💡Examples
Problem 1:
A solid is formed by placing 4 cubes of side unit each in a 'T' shape on a table (three cubes in a row and one cube attached to the middle cube at the front). Describe the Top view and the Front view.
Solution:
- Top View: Looking from above, we see all 4 cubes arranged. The three cubes in a row form a horizontal block, and the fourth cube extends out from the center. This creates a 'T' shape on the grid.
- Front View: Looking from the front of the single protruding cube, we see that cube in the foreground. However, because orthographic views project onto a plane, the three cubes in the back row are also visible. The view appears as a horizontal rectangle of squares, assuming the height is uniform.
Explanation:
To find the views, imagine projecting the 3D structure onto 2D planes located above and in front of the object. In the top view, the 'depth' of the T is visible. In the front view, the depth is flattened into a single plane.
Problem 2:
Verify Euler's Formula for a Square Pyramid. A square pyramid has faces and vertices.
Solution:
Step 1: Identify the components. Given (1 square base + 4 triangular faces) and (4 base corners + 1 apex). Step 2: Identify the number of edges (). A square pyramid has edges on the base and edges meeting at the apex, so . Step 3: Substitute into Euler's Formula . Since , Euler's formula is verified.
Explanation:
Euler's Formula applies to all convex polyhedrons. By counting the faces, vertices, and edges and plugging them into the equation, we can verify the topological properties of the solid.