Mensuration - Area of Rhombus

A rhombus is a quadrilateral where all four sides are of equal length ($s$). Visually, it looks like a diamond or a slanted square. Unlike a square, its internal angles do not have to be $90^{\circ}$, but opposite angles are always equal.

The diagonals of a rhombus, denoted as $d_1$ and $d_2$, connect opposite vertices. A key visual property is that these diagonals are perpendicular to each other, crossing at a $90^{\circ}$ angle to form a cross shape inside the figure.

The diagonals of a rhombus bisect each other at right angles. This means they cut each other exactly in half. Visually, the intersection point serves as the midpoint for both diagonal lines.

The Area of a rhombus represents the total space enclosed within its four sides. If the lengths of the diagonals are known, the area is calculated as half their product: $Area = \\frac{1}{2} \\times d_1 \\times d_2$. Visually, this area is exactly half of the rectangle formed by the lengths of the diagonals.

Because every rhombus is a parallelogram, the area can also be found using a side as the base and multiplying it by the altitude ($h$). The altitude is the perpendicular distance or the 'height' between two opposite parallel sides.

The Perimeter of a rhombus is the total length of its outer boundary. Since all four sides are equal, it is visually and mathematically calculated as $4 \\times s$, where $s$ is the length of one side.

The relationship between the side and the diagonals can be visualized using the Pythagorean theorem. The diagonals divide the rhombus into four identical right-angled triangles. In each triangle, the side $s$ is the hypotenuse, meaning $s = \\sqrt{(\\frac{d_1}{2})^2 + (\\frac{d_2}{2})^2}$.