Diagonal of Rhombus — Formula, Properties, Examples

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.

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