Area of a Rhombus

Area of a Rhombus

How to find the area of a rhombus: formula, proof, examples, and their solutions.

Formula

A = (1/2)*bh. A: area of a rhombus. a, b: diagonals

A = (1/2)⋅ab

A: area of a rhombus
a, b: diagonals

Proof

Area of a Rhombus: Proof of the Formula

The diagonals of a rhombus
perpendicularly bisect each other.

Then the gray right triangle's legs are
(1/2)a and (1/2)b.

So the area of the gray right triangle is (1/8)ab.

There are 4 right triangles in the rhombus.

So the area of the rhombus is
4 × (1/8)ab = (1/2)ab.

Example 1

Find the area of the given rhombus. AC = 12, BD = 7.

a = 7, b = 12

A = (1/2)⋅7⋅12

Example 2

Find the area of the given rhombus. Segment from bisected diagonal: 4, Side: 5.

See the darker gray right triangle.
Its hypotenuse is 5.
And its leg is 4.

So it's a (3, 4, 5) triangle.

So the green leg is 3.

The diagonals perpendicularly bisect each other.
So the blue segments are 4.
And the green segments are 3.

So a = 8, b = 6.

a = 8, b = 6

A = (1/2)⋅8⋅6