Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
Introduction to 2D Views: 3D objects occupy space and have three dimensions: length, breadth, and height. When we look at these objects from different positions—Top, Front, and Side—they appear as 2D shapes. For example, a cylinder viewed from the top looks like a circle, but from the front or side, it looks like a rectangle.
Slicing and Cross-sections: Slicing a solid involves cutting it with a sharp tool to reveal an internal surface. This 2D surface is called a 'cross-section'. Imagine a long cucumber; a vertical cut (perpendicular to its length) produces a circular cross-section, while a horizontal cut (parallel to its length) produces a rectangular or oval cross-section.
Vertical and Horizontal Cuts: The orientation of a cut determines the shape of the cross-section. For a square pyramid, a vertical cut passing through the apex (top point) results in a triangular cross-section, whereas a horizontal cut parallel to the base results in a square cross-section that is smaller than the actual base.
Shadow Play: When a 3D object is placed in the path of light, it casts a 2D shadow on a screen or floor. The shape of the shadow depends on the orientation of the object relative to the light source. A sphere always casts a circular shadow, but a cube can cast a square shadow or even a hexagonal shadow depending on how it is held against the light.
Direction of Light and Shadow Shape: If a light source (like a torch) is held directly above a cylindrical pipe, the shadow on the floor is a rectangle. However, if the torch is shone at the circular end of the pipe, the shadow formed on the wall behind it is a circle.
Visualising Solid Sections: To visualize the section of a solid, one must imagine the 'hidden' faces and how the cutting plane intersects the edges. For instance, slicing a cone horizontally results in a circle, but slicing it at an angle (without hitting the base) results in an ellipse, which looks like a stretched circle.
Orthographic Projections: This is a method of representing 3D objects in 2D by projecting views from different directions. When viewing a stack of identical cubes, the 'Top View' shows the layout of the cubes on the ground, while the 'Front View' shows the maximum height and width of the arrangement.
📐Formulae
Euler's Formula for Polyhedrons: , where is the number of faces, is the number of vertices, and is the number of edges.
Area of a Circular Cross-section:
Area of a Rectangular Cross-section:
💡Examples
Problem 1:
What cross-section do you get when you give a (i) vertical cut and (ii) horizontal cut to a brick?
Solution:
(i) A vertical cut (perpendicular to the base) of a brick results in a rectangular cross-section. (ii) A horizontal cut (parallel to the base) of a brick also results in a rectangular cross-section.
Explanation:
A brick is a cuboid. Since all faces of a cuboid are rectangles, any cut made parallel to any of its faces will result in a rectangular cross-section. The dimensions of the rectangle will depend on the dimensions of the brick along the plane of the cut.
Problem 2:
A light bulb is placed directly above a circular cylinder which is standing vertically on its base. What is the shape of the shadow obtained on the ground?
Solution:
The shape of the shadow will be a circle.
Explanation:
When the light source is directly above the cylinder, the rays are blocked by the top circular face. The projection of this circular face onto the horizontal ground results in a 2D circular shadow. If the light were coming from the side, the shadow would have been a rectangle.