How Diagonals Relate to Side Length and Angles in a Rhombus

The diagonals of a rhombus are not just lines connecting corners; they hold the key to its unique geometry, defining its angles and its relationship with its side length.

Key Properties of Rhombus Diagonals

• They are Perpendicular Bisectors: This is the most important property. The two diagonals (`d₁` and `d₂`) cross each other at their midpoints, forming a perfect 90-degree angle (right angle).
• They Bisect the Angles: Each diagonal cuts the internal angles at its vertices exactly in half. For example, if a rhombus has angles of 100° and 80°, one diagonal will split the 100° angles into two 50° angles, and the other diagonal will split the 80° angles into two 40° angles.
• They are Not (Usually) Equal: Unlike a square, the two diagonals of a rhombus are not equal in length (unless the rhombus is also a square). One diagonal is longer, and one is shorter.

Relationship Between Diagonals and Side Length

This relationship is derived using the Pythagorean theorem.

1. Because the diagonals are perpendicular bisectors, they divide the rhombus into four identical right-angled triangles.
2. For any of these right-angled triangles:
   • The hypotenuse (the longest side) is the side of the rhombus (s).
   • The two shorter legs are half of each diagonal: `d₁ / 2` and `d₂ / 2`.

According to the Pythagorean theorem (`a² + b² = c²`), we get the fundamental rhombus formula:

(d₁ / 2)² + (d₂ / 2)² = s²

This formula allows you to find the side length (s) if you know the diagonals, or find one diagonal if you know the side length and the other diagonal.

Relationship Between Diagonals and Angles

Using the same right-angled triangles, we can use trigonometry (SOH CAH TOA) to find the internal angles of the rhombus if we know the diagonal lengths.

• Let `θ` be one of the half-angles at a vertex.
• In the right-angled triangle, `tan(θ) = (opposite / adjacent)`.
• `tan(θ) = ( (d₁ / 2) / (d₂ / 2) ) = d₁ / d₂`

By finding the half-angle `θ` using the inverse tangent `(arctan(d₁/d₂))`, you can then double it to find the full internal angle of the rhombus.