The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once. For a polygon (a closed plane figure made up of line segments), the perimeter is the sum of the lengths of all its sides.
$$\text{Perimeter of a polygon} = \text{Sum of the lengths of all its sides}$$
Opposite sides of a rectangle are always equal in length. Therefore, the perimeter of a rectangle is twice the sum of its length and breadth:
$$\text{Perimeter of a rectangle} = 2 \times (\text{length} + \text{breadth})$$
Example: Consider a rectangle with length $12\text{ cm}$ and breadth $8\text{ cm}$:
$$\text{Perimeter} = 2 \times (12\text{ cm} + 8\text{ cm}) = 2 \times 20\text{ cm} = 40\text{ cm}$$
Since all four sides of a square are equal in length, the perimeter is four times the length of one side:
$$\text{Perimeter of a square} = 4 \times \text{length of a side}$$
Example: For a square photo frame of side $1\text{ m}$:
$$\text{Perimeter} = 4 \times 1\text{ m} = 4\text{ m}$$
The perimeter of a triangle is the sum of the lengths of its three sides:
$$\text{Perimeter of a triangle} = a + b + c$$
Example: For a triangle with sides $4\text{ cm}$, $5\text{ cm}$, and $7\text{ cm}$:
$$\text{Perimeter} = 4\text{ cm} + 5\text{ cm} + 7\text{ cm} = 16\text{ cm}$$
Closed figures that have all sides and all angles equal are called regular polygons.
$$\text{Perimeter of an equilateral triangle} = 3 \times \text{length of a side}$$
$$\text{Perimeter of a regular polygon} = \text{Number of sides} \times \text{Length of a side}$$
When measuring perimeters on a grid, we can distinguish between two types of unit lengths:
The perimeter of a figure can be expressed in terms of these units, such as $6s + 3d$ units.
When a shape is cut and its pieces are rearranged to form new shapes, the perimeter can change even though the area remains the same.
For example, a rectangular paper of $6\text{ cm} \times 4\text{ cm}$ cut into two equal pieces can be joined to form different shapes with perimeters like $28\text{ cm}$ or $22\text{ cm}$.
The amount of region enclosed by a closed figure is called its area.
$$\text{Area of a square} = \text{side} \times \text{side}$$
$$\text{Area of a rectangle} = \text{length} \times \text{breadth}$$
Example: A floor is $5\text{ m}$ long and $4\text{ m}$ wide. A square carpet of side $3\text{ m}$ is laid on the floor. Find the area of the floor that is not carpeted.
To estimate the area of any simple closed shape on a squared or graph paper (where each square is $1\text{ unit} \times 1\text{ unit}$), follow these conventions:
Squares are the standard unit for measuring area because they can be packed tightly together without leaving any gaps or overlaps. Other shapes, like circles, leave gaps when packed together, making them unsuitable for accurate area measurement.
The relationship between the area of a rectangle and a triangle can be understood by drawing a diagonal:
$$\text{Area of each triangle} = \frac{1}{2} \times \text{Area of the rectangle} = \frac{1}{2} \times (\text{length} \times \text{breadth})$$
This principle also applies when dividing a rectangle into multiple parts, where the sum of the areas of the triangles equals half the sum of the areas of the corresponding rectangles.