Mensuration - Area of Trapezium and General Quadrilaterals

A trapezium is a quadrilateral with one pair of parallel sides. Visually, imagine a four-sided figure where the top side $a$ and the bottom side $b$ are parallel lines, and the vertical gap between them represents the height $h$.

The area of a trapezium is calculated as half the product of the sum of the parallel sides and the perpendicular distance between them. This formula essentially takes the average of the two parallel bases and multiplies it by the height.

A general quadrilateral can be divided into two triangles by drawing one of its diagonals. Visually, imagine a diagonal $d$ splitting the shape; we then draw two perpendicular lines (heights $h_1$ and $h_2$) from the opposite vertices to this diagonal.

The area of a general quadrilateral is the sum of the areas of the two triangles formed by the diagonal. This is expressed as $\frac{1}{2} \times d \times (h_1 + h_2)$, where $d$ is the diagonal and $h_1, h_2$ are the perpendicular offsets.

A rhombus is a special quadrilateral where all sides are equal and diagonals bisect each other at right angles. Visually, it resembles a diamond shape where the two internal diagonals $d_1$ and $d_2$ cross at a $90^{\circ}$ angle.

The area of a rhombus can be found using its diagonals. Since the diagonals are perpendicular, the area is simply half the product of the lengths of the two diagonals.

The height or altitude of any quadrilateral is the perpendicular distance from a vertex to the opposite side or between two parallel sides. In geometry diagrams, this is denoted by a line meeting a base at a $90^{\circ}$ angle, often marked with a small square symbol.