How to Derive Area Formulas for a Rhombus

A rhombus is a quadrilateral with all four sides of equal length. There are two primary formulas used to find its area, and both can be easily derived from its geometric properties.

Method 1: Using Base and Height (Like a Parallelogram)

A rhombus is a special type of parallelogram. Therefore, we can use the standard area formula for a parallelogram:

Area = base × height

• Base (b): The length of any one of its four equal sides.
• Height (h): The perpendicular distance between the chosen base and the opposite side.

Derivation: If you "snip" a right-angled triangle from one side of the rhombus and move it to the other, it forms a perfect rectangle with a length of 'base' and a width of 'height'. The area of this new rectangle is `base × height`, which is the same as the area of the original rhombus.

Method 2: Using the Diagonals

This is the most common method specific to a rhombus. The diagonals of a rhombus (let's call them `d₁` and `d₂`) have a special property: they bisect each other at a perfect 90-degree angle (right angle).

Area = (d₁ × d₂) / 2

Derivation:

1. The two diagonals (`d₁` and `d₂`) divide the rhombus into four congruent right-angled triangles.
2. Let's look at just one of these triangles. Its two legs (the sides that form the right angle) are half the length of each diagonal.
   • Leg 1 = `d₁ / 2`
   • Leg 2 = `d₂ / 2`
3. The area of a right-angled triangle is `(leg 1 × leg 2) / 2`. So, the area of one small triangle is:
   Area_triangle = `( (d₁ / 2) × (d₂ / 2) ) / 2` = `(d₁ × d₂) / 8`
4. Since there are four of these identical triangles in the rhombus, the total area is:
   Total Area = 4 × Area_triangle
   Total Area = 4 × `( (d₁ × d₂) / 8 )`
   Total Area = `(4 × d₁ × d₂) / 8`
   Total Area = `(d₁ × d₂) / 2`

This formula, `Area = (d₁ × d₂) / 2`, is a very efficient way to find the area if the diagonal lengths are known.