Mensuration - Perimeter and Area of Squares and Rectangles

Perimeter is the total length of the outer boundary of a closed 2-dimensional shape. Visualise it as the total length of a string needed to wrap around the edges of the object. For a rectangle, it is the sum of all four sides: $l + b + l + b$, or $2(l + b)$. For a square, it is $s + s + s + s$, or $4s$.

Area measures the surface region enclosed within the boundaries of a shape. Imagine a floor covered with equal-sized square tiles; the count of these tiles represents the area. For a rectangle, the area is the product of its length and breadth ($l \\times b$). For a square, it is the side multiplied by itself ($s^2$).

A Rectangle is a quadrilateral where opposite sides are equal and parallel. All four interior angles are right angles (90^\\circ). The longer dimension is called the length ($l$) and the shorter dimension is the breadth ($b$).

A Square is a special type of rectangle where all four sides are equal in length ($s$). Because all sides are equal, the area and perimeter formulas are simplified versions of the rectangle's formulas.

The Diagonal is the straight line connecting two opposite corners of the shape. In a rectangle, the diagonal $d$ forms a right-angled triangle with the length and breadth, which allows us to use the Pythagorean theorem: $d = \\sqrt{l^2 + b^2}$. In a square, the diagonal is $d = s\\sqrt{2}$.

Units of Measurement: Perimeter is measured in linear units like $\\text{cm}$, $\\text{m}$, or $\\text{km}$. Area is measured in square units like $\\text{cm}^2$, $\\text{m}^2$, or $\\text{km}^2$. Always convert all dimensions to the same unit before starting a calculation.

Practical Applications: Fencing a field involves calculating the perimeter, as it relates to the boundary. Flooring, carpeting, or painting a wall involves calculating the area, as it relates to the surface region.

Relationship between Perimeter and Area: Two different rectangles can have the same perimeter but different areas, or the same area but different perimeters. However, for a given perimeter, a square will always enclose the maximum possible area.