# Diagonal of Rhombus — Formula, Properties, Examples

TL;DR

The two diagonals of a rhombus bisect each other at right angles, splitting it into four congruent right triangles. That single fact gives you every diagonal formula: from area, p=2×Area/q, and from the side, (p/2)²+(q/2)²=a². This guide derives both and works through six examples.

## What Are The Diagonals of A Rhombus?

A rhombus is a quadrilateral with all four sides equal in length. Its **diagonals** are the two line segments that join opposite corners, and unlike in a general parallelogram, a rhombus's diagonals cross each other at **90°** and each one cuts the other exactly in half.

Call the diagonals p and q, and they meet at the rhombus's centre. Because they bisect each other, the centre splits p into two halves of p/2 and q into two halves of q/2. Because they meet at a right angle, those halves are the legs of four identical right triangles, and almost every diagonal calculation comes back to one of those triangles.

## Properties of The Diagonals of A Rhombus

Three properties do all the work, and each one is worth stating before any formula.

- **The diagonals bisect each other.** Like every parallelogram, a rhombus has diagonals that share a midpoint.
- **The diagonals are perpendicular.** They meet at exactly 90°.
- **The diagonals bisect the vertex angles.** Each diagonal cuts the two corner angles it reaches into two equal parts.

Put the first two together and you get the picture that powers everything: the diagonals divide the rhombus into **four congruent right-angled triangles**.

## Diagonal of Rhombus Formula

There are two formulas, depending on what you already know. Each symbol: p and q are the two diagonals, a is the side length, and Area is the rhombus's area.

**When you know the area and one diagonal:** p=2×Area/q

**When you know the side and one diagonal (from the right triangle):** (p/2)²+(q/2)²=a²

The first comes from the area formula; the second comes from the Pythagorean theorem.

## How Is The Diagonal Formula Derived?

### From the area

The area of a rhombus is half the product of its diagonals: Area=1/2 pq. This itself comes from the four right triangles, each has area 1/2 * (p/2) * (q/2), which gives 4 * (1/2 * (p/2) * (q/2)) = 1/2 pq. Rearranging gives:

p=2×Area/q.

### From the side using Pythagoras

Take one of the four right triangles. Its legs are p/2 and q/2, and its hypotenuse is a. The Pythagorean theorem gives:

(p/2)²+(q/2)²=a².

Multiply through by 4 and it tidies into:

p² + q² = 4a².

This is the relationship to reach for whenever a problem gives you the side and one diagonal and asks for the other.

## Examples of Diagonal of Rhombus

### Example 1

**A rhombus has an area of 54 cm² and one diagonal measuring 6 cm. Find the other diagonal.**

p=2×54/6=18 cm

**Final answer:** The other diagonal is 18 cm.

### Example 2

**A rhombus has a side of 13 cm and one diagonal of 10 cm.**

First instinct: The side is the hypotenuse, half of the known diagonal is 5, let the other half-diagonal be q/2.

5²+(q/2)²=13²

The correct method gives:

**Final answer:** The other diagonal is 24 cm, not 3 cm.

### Example 3

**The diagonals of a rhombus are 16 cm and 30 cm. Find the side length.**

Half-diagonals are 8 cm and 15 cm; the side is:

a²=8²+15² → a=17 cm

**Final answer:** The side is 17 cm.

### Example 4

**A rhombus has diagonals of 10 cm and 24 cm. Find its area.**

Area=1/2×10×24=120 cm²

### Example 5

**A rhombus has a perimeter of 40 cm and one diagonal of 12 cm.**

First find the side from the perimeter.
a=40/4=10 cm

Now use the right triangle:

6²+(q/2)²=10² → **Final answer:** The other diagonal is 16 cm.

### Example 6

**A rhombus has diagonals of 14 cm and 48 cm. Find its side, then its area.**

Half-diagonals are 7 cm and 24 cm:

a²=7²+24² → a=25 cm

Area=1/2×14×48=336 cm²

**Final answer:** Side = 25 cm, Area = 336 cm².

## Why The Diagonals of A Rhombus Matter

The perpendicular diagonals of a rhombus are essential in understanding load distribution in structures, such as car jacks and trusses. This visual representation aids in applying the Pythagorean theorem beyond simple triangles.

## Tripping Points To Avoid

### Mistake 1: Forgetting to halve the diagonals before using Pythagoras

### Mistake 2: Assuming the diagonals are equal

### Mistake 3: Using the side as the area shortcut

## Conclusion

- The **diagonals of a rhombus** bisect each other at 90°, forming four congruent right triangles.
- They are generally unequal; equal diagonals would make it a square.
- The diagonal formulas are derived from both area and side length, emphasizing the importance of using half-diagonals in triangle applications.