Mathematics: Class 5 — Chapter 11: Grandmother’s Quilt

11.1 Introduction to Perimeter

The boundary or border of any closed shape has a specific length. This length of the border of a shape is called its perimeter.

Perimeter of Regular Shapes: If a shape has all equal sides, its perimeter can be found by multiplying the length of one side by the total number of sides.
Example 1: A regular pentagon (5 sides) with each side measuring $4\text{ cm}$ has a perimeter of: $$\text{Perimeter} = 5 \times 4\text{ cm} = 20\text{ cm}$$
Example 2: An hourglass-like shape made of two equilateral triangles sharing a vertex, with each of the 6 outer boundary sides measuring $5\text{ cm}$, has a perimeter of: $$\text{Perimeter} = 6 \times 5\text{ cm} = 30\text{ cm}$$

11.2 Understanding Area and Tiling

The region covered by a shape on a flat surface is called its area.

To find the area of a region, we fill it with smaller shapes that tile (leave no gaps and no overlaps), such as squares, rectangles, and triangles.
Do circles tile? No, circles cannot tile a flat surface completely because they always leave gaps when placed side-by-side.
The area of a surface can be measured in terms of the number of unit shapes (e.g., triangle units, square units, rectangle units) required to cover it completely.

11.3 Measuring Area with Square Grids

Using square grids is the most convenient and standard way to find the area of different regions.

The area of a square with sides of $1\text{ unit}$ is defined as 1 unit square (or $1\text{ square cm}$ if the side is $1\text{ cm}$).
By placing shapes on a grid of $1\text{ cm} \times 1\text{ cm}$ squares, we can count the number of grid squares enclosed by the shape to determine its area.

11.4 Relationship Between Area and Perimeter

Area and perimeter are two different measures of a shape. An increase in area does not always mean an increase in perimeter, and vice versa.

11.5 Formulas for Rectangles and Squares

Instead of counting individual grid squares, we can use mathematical formulas to calculate the area and perimeter of rectangles and squares.

Area of a Rectangle: $$\text{Area} = \text{Length} \times \text{Breadth}$$
Perimeter of a Rectangle: $$\text{Perimeter} = 2 \times \text{Length} + 2 \times \text{Breadth} = 2 \times (\text{Length} + \text{Breadth})$$

11.6 Practice Problems

Problem 1: Find the area of a rectangular field whose length is $42\text{ m}$ and breadth is $34\text{ m}$.
Solution: $\text{Area} = 42\text{ m} \times 34\text{ m} = 1428\text{ square m}$.
Problem 2: The area of a rectangular garden is $64\text{ square m}$ and its length is $16\text{ m}$. What is its breadth?
Solution: $\text{Breadth} = \frac{\text{Area}}{\text{Length}} = \frac{64}{16} = 4\text{ m}$.