Mensuration - Area of Polygons

Definition of Area: Area is the measure of the region enclosed within the boundary of a closed plane figure. It is expressed in square units such as $cm^{2}$ or $m^{2}$. Visually, it represents the number of unit squares that can fit inside the shape.

Area of a Trapezium: A trapezium is a quadrilateral with exactly one pair of parallel sides. Visually, imagine a shape with a top horizontal line ($a$) and a longer bottom horizontal line ($b$), connected by two slanted side legs. The area is found by multiplying the average of the parallel sides by the perpendicular distance ($h$) between them.

Area of a General Quadrilateral: Any irregular four-sided polygon can be divided into two triangles by drawing a diagonal ($d$). Visually, if you draw a line from one corner to the opposite corner, you create two triangles. The area of the quadrilateral is the sum of the areas of these two triangles, calculated using the diagonal as a base and the heights ($h_1$ and $h_2$) dropped from the other two vertices.

Area of a Rhombus: A rhombus is a quadrilateral where all sides are equal. A unique property is that its diagonals ($d_1$ and $d_2$) intersect at right angles ($90^{\circ}$) and bisect each other. Visually, it looks like a tilted square where the area is half the product of the lengths of these internal perpendicular diagonals.

Area of a Parallelogram: A parallelogram is a quadrilateral where opposite sides are parallel and equal. Visually, it resembles a rectangle that has been pushed to one side. Its area is determined by the length of its base ($b$) and the vertical height ($h$) perpendicular to that base.

Area of a General Polygon: Complex polygons (like pentagons or hexagons) can be measured by dividing them into simpler shapes such as triangles, rectangles, or trapezia. Visually, this is done by drawing internal diagonals or perpendicular lines to a central base line, effectively turning one large irregular shape into a 'puzzle' of smaller known shapes.

Relationship between Units: When calculating area, all dimensions must be in the same unit. Remember that $1$ $cm^{2} = 100$ $mm^{2}$ and $1$ $m^{2} = 10,000$ $cm^{2}$. In a visual context, $1$ $m^{2}$ is a square with side lengths of $100$ $cm$ each.