Measurement - Area of Squares, Rectangles, Triangles, and Parallelograms

Area is the measure of the space inside a two-dimensional shape, calculated in square units such as $cm^2$ or $m^2$. Visually, imagine covering the surface of a shape with small $1 \times 1$ squares; the area is the total number of those squares required to fill the shape completely.

For a Rectangle, the area is the product of its length and width ($l \times w$). You can visualize this as a grid where the length represents the number of unit squares in one horizontal row and the width represents how many of those rows are stacked vertically.

A Square is a specific type of rectangle where all four sides are equal length ($s$). The area is $s \times s$ or $s^2$. Visually, this creates a perfectly symmetrical grid with an identical number of squares along both the base and the side.

A Parallelogram's area is calculated using the base and the perpendicular height ($b \times h$). To visualize why this works, imagine cutting a right-angled triangle from one side of the parallelogram and sliding it to the opposite side. This transformation turns the parallelogram into a rectangle with the same base and height.

The Height ($h$) of a shape is the perpendicular distance from the base to the highest point. In diagrams, this is often represented by a dashed vertical line forming a $90^{\circ}$ angle (indicated by a small square symbol) with the base. It is important to distinguish this from the 'slant height' or the length of the diagonal side.

The Area of a Triangle is exactly half the area of a parallelogram with the same base and height, leading to the formula $\frac{1}{2} \times b \times h$. Visually, if you take any triangle and attach an identical copy rotated $180^{\circ}$ along one side, the two triangles will form a parallelogram.

Before performing calculations, ensure that all units of measurement are consistent. If a rectangle has a length in $cm$ and a width in $mm$, you must convert them to the same unit first so that the resulting area is in a standard square unit like $cm^2$.