Mensuration - Area of Polygons

Polygon Definition and Classification: A polygon is a closed plane figure bounded by three or more line segments. A 'Regular Polygon' has all sides and all interior angles equal (like an equilateral triangle or a square), while an 'Irregular Polygon' has sides or angles of different measures. Visually, a regular polygon looks perfectly symmetrical around its center.

Area of a Trapezium: A trapezium is a quadrilateral with exactly one pair of parallel sides. To visualize its area, imagine a rectangle formed by the average length of the two parallel sides ($a$ and $b$) and the perpendicular distance ($h$) between them. The formula calculates the area as half the product of the sum of parallel sides and the height.

Area of a General Quadrilateral: Any general quadrilateral can be viewed as two triangles sharing a common diagonal. Visually, if you draw a diagonal $d$ and then draw perpendiculars (called offsets) $h_1$ and $h_2$ from the remaining two vertices to this diagonal, the area of the quadrilateral is the sum of the areas of these two triangles.

Area of a Rhombus: A rhombus is a special parallelogram where all sides are equal. Visually, its diagonals ($d_1$ and $d_2$) intersect at right angles ($90^{\circ}$) and bisect each other. The area can be found using the lengths of these diagonals or by treating it as a parallelogram with a base and a height.

Area of a Parallelogram: A parallelogram has opposite sides that are parallel and equal. Visually, if you cut a right-angled triangle from one side and move it to the other, it forms a rectangle. Thus, the area is simply the product of its base length and its corresponding vertical height.

Area of Irregular Polygons (Field Book Method): To find the area of an irregular polygon or a field, we divide the entire shape into several smaller, non-overlapping known figures such as triangles and trapezia. Visually, this involves drawing a central baseline and measuring perpendicular offsets to various vertices, then summing the individual areas of the resulting sub-shapes.

Area of a Regular Hexagon: A regular hexagon consists of six congruent equilateral triangles meeting at a central point. Visually, if you draw lines from the center to each vertex, you see six identical triangles. The total area is $6$ times the area of one equilateral triangle with side $s$.