NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 6: Mensuration - Multiple Choice Questions

Question 1

Why (ii) has the smallest perimeter

1. Perimeter means the total length of the outer boundary.

2. Each unit square has 4 sides. For 6 unit squares, total sides before joining is:

   \(4 \times 6 = 24\)

   (This counts all sides, even the ones that will be shared inside.)

3. When two squares touch along a full side, that side is shared. A shared side is not on the outside, so it reduces the perimeter by 2 (because we counted it for both squares).

   So, more shared sides ⇒ smaller perimeter.

4. Among the given figures, (ii) is the most compact: it is a 2 × 3 rectangle. Compact shapes have the maximum number of shared sides.

5. Count shared sides in the 2 × 3 rectangle:

   • Horizontal shares: 2 rows, each row has \(3 - 1 = 2\) shared sides.

     So, \(2 \times 2 = 4\) shared sides.

   • Vertical shares: 3 columns, each column has \(2 - 1 = 1\) shared side.

     So, \(3 \times 1 = 3\) shared sides.

   Total shared sides = \(4 + 3 = 7\).

6. Now compute the perimeter:

   Start with 24 sides.

   Each shared side removes 2 from the perimeter count.

   So, perimeter = \(24 - 2 \times 7\)

   = \(24 - 14\)

   = \(10\) units.

7. Any less compact shape (like (i), (iii), (iv)) has fewer shared sides, so its perimeter will be more than 10 units.

Therefore, figure (ii) (the 2 × 3 rectangle) has the smallest perimeter.