Visualising Solid Shapes: NCERT Class 7 Maths Exercise 15.1

What is a 2D Net of a 3D Solid?

To visualize a solid shape, imagine peeling it open or cutting it along some edges and laying it flat on a table. The resulting flat 2D pattern is called a net of that solid shape. For example, if you unfold a cardboard box, you get a flat arrangement of six squares or rectangles. Not every layout of six squares can be folded to form a cube. To be a valid net, the faces must fold along their shared borders without overlapping, eventually closing up perfectly to enclose a 3D space. Exercise 15.1 challenges you to train your mind's eye to visually fold these flat nets, checking if they successfully form closed 3D solids like cubes, cylinders, cones, or pyramids.

How to Verify If a Net Can Form a Cube

1. Count the Faces — Ensure the flat net has exactly 6 squares. A cube has 6 faces. If a net has 5 or 7 squares, it can never form a complete, single-layered cube.
2. Identify the Base and Fold Imaginary Walls — Choose one central square to be the 'bottom' or base face. Mentally fold the adjacent squares 90 degrees upward to form the side walls.
3. Fold the Lid — Check if the remaining squares can fold over to form the top 'lid' of the cube without colliding or overlapping with an already established side.

The Golden Dice Rule

A standard dice is a cube where the dots on opposite faces always add up to exactly 7. In CBSE Class 7 exams, questions often ask you to fill in missing numbers on a folded dice net. To solve these easily:

1. Identify which squares will become opposite faces when folded.
2. In a flat strip of 3 or 4 squares, alternate squares (separated by exactly one square) always fold to become opposite faces.
3. Ensure that the sum of these pairs equals 7!

Practice Questions with Solutions

• Q: Can a net with 6 squares arranged in a straight T-shape (with 4 vertical squares and 2 side wings attached to the second square) fold into a cube? A: Step 1: Count the faces. There are exactly 6 squares. Step 2: Let's choose the second square from the top of the vertical line as the base. Step 3: Fold the top square up to form the back wall, and fold the bottom two squares to form the front wall and the top lid. Step 4: Fold the two side wings upward to form the left and right walls. Final answer: Yes, this T-shaped net can be folded into a complete cube without any overlap.
• Q: A dice net has three visible face numbers: 1, 2, and 3. If 1 is opposite to 6, 2 is opposite to 5, and 3 is opposite to 4, construct the missing opposite face values if you are given a horizontal strip of four squares numbered [1, 2, 6, x] with two wings [3] and [y] attached. Find the values of x and y. A: Step 1: In a flat strip of four squares [1, 2, 6, x], alternate squares become opposite faces. Thus, the 1st square (1) is opposite to the 3rd square (6). This fits the standard dice rule: 1 + 6 = 7. Step 2: The 2nd square (2) is opposite to the 4th square (x). For a standard dice, opposite faces must sum to 7. So, 2 + x = 7, which gives x = 5. Step 3: The side wings always fold to oppose each other. Thus, wing [3] must be opposite wing [y]. Under the dice rule, 3 + y = 7, which gives y = 4. Final answer: The missing face values are x = 5 and y = 4.
• Q: Match the 3D solid with its appropriate net: 1. Cylinder 2. Cone (a) One sector of a circle and a circular base. (b) One rectangle and two circular bases. A: Step 1: Analyze a cylinder. Unfolding a cylinder vertically along its height gives a single flat rectangular curved surface, with two flat circular bases at either end. This matches description (b). Step 2: Analyze a cone. Unfolding a cone along its slant height gives a pie-slice shaped circle sector (for the curved surface) and a single flat circular base. This matches description (a). Final answer: 1 matches with (b), 2 matches with (a).
• Q: Why can a straight strip of six squares in a single line never fold into a cube? A: Step 1: Consider folding a strip of 6 squares in a single continuous line. Step 2: When you fold along the lines, the 1st, 2nd, 3rd, and 4th squares fold around to form the four sides of a loop (lateral faces). Step 3: The 5th and 6th squares will continue to fold around this same loop, overlapping directly with the 1st and 2nd faces. Final answer: Since there are no lateral side flaps to close the left and right open ends, the straight strip forms an open tube with double-layered sides, not a closed cube.

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