Visualising Solid Shapes Class 7 NCERT Exercise 15.3

What is a Cross-Section?

A cross-section is the 2D shape you get when you cut straight through a 3D solid. Imagine taking a sharp kitchen knife and slicing through a solid object. The flat surface that is revealed is our cross-section.

There are two main ways to cut a solid shape relative to its base:

1. Vertical Cut (Perpendicular to the base): Cutting straight down from the top to the bottom. For example, if you cut a loaf of bread vertically, you get a rectangular-shaped slice.
2. Horizontal Cut (Parallel to the base): Cutting sideways, parallel to the flat ground. For instance, if you cut a vertical carrot horizontally, you get circular discs.

The shape of the cross-section depends entirely on the direction of the cut and the geometry of the original 3D solid.

Step-by-Step Guide to Visualising Cross-Sections

1. Identify the Solid Shape — Look closely at the 3D solid. Is it a prism, a cylinder, a sphere, a cone, or a cuboid? Note down its general features like curved or flat faces.
2. Determine the Cut Direction — Identify if you are making a vertical cut (top-to-bottom, perpendicular to the base) or a horizontal cut (left-to-right, parallel to the base).
3. Sketch the Boundary Plane — Imagine a flat sheet of paper passing through the solid along the cut. Trace the lines where the sheet touches the boundary of the solid.
4. Name the 2D Shape — Look at the enclosed boundary face on your slice. Name the 2D shape (like a circle, square, triangle, or rectangle).

Slicing Common Solid Shapes

Slicing Visualisation Secrets for Exams

• Spheres are Unique: No matter which direction you slice a perfect sphere (vertical, horizontal, or slanted), the cross-section is always a Circle!
• Watch the Cylinder: A vertical slice of a cylinder standing on its base does not make a circle; it reveals a Rectangle. This is a very common exam trap!
• Cone Slicing: A vertical cut exactly through the topmost point (vertex) of a cone results in a Triangle.

Practice Questions with Solutions

• Q: What cross-sections do you get when you give a (i) vertical cut and (ii) horizontal cut to a die (cube)? A: Step 1: Recall that a die is a perfect Cube. All its faces are equal squares. Step 2: For a vertical cut, slicing straight down from the top face parallel to the side faces yields a square. Step 3: For a horizontal cut, slicing parallel to the base also yields a square. Final answer: (i) Square, (ii) Square.
• Q: A round pipe made of metal is placed vertically. What cross-section is obtained from (i) a vertical cut and (ii) a horizontal cut? A: Step 1: A round metal pipe is cylindrical in shape. Step 2: If we slice a cylinder vertically (from top to bottom), the cross-section is a flat rectangular strip. Step 3: If we slice the vertical cylinder horizontally (parallel to the base), the cross-section is a circle. Final answer: (i) Rectangle, (ii) Circle.
• Q: What is the shape of the cross-section when you make a horizontal cut through an ice-cream cone standing on its tip? A: Step 1: An ice-cream cone is a Cone. Step 2: The base of a cone is a circle. Step 3: Any horizontal cut made parallel to the circular base of a cone will produce a circular cross-section (of smaller size than the base). Final answer: Circle.
• Q: Imagine cutting a cuboidal wooden block vertically. Describe the shape of the cross-section and explain why. A: Step 1: A wooden block is a cuboid. Its faces are rectangles. Step 2: When sliced vertically, the cut runs parallel to the side faces of the cuboid. Step 3: Therefore, the resulting face exposed is a rectangle, matching the cross-section dimensions. Final answer: Rectangle.

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