Rhombus geometry

A rhombus is a parallelogram with four equal side lengths. You can think of it as a square that’s been “tilted” or skewed. Some properties match a square, but others don’t. This section reviews the key properties and how to find a rhombus’s area.

Geometric properties

Since a rhombus is a parallelogram, it has the same basic parallelogram properties:

• Opposing sides are parallel. The top and bottom sides are parallel, and the left and right sides are parallel.

Because all four sides are equal, a rhombus also has angle relationships that come from being a parallelogram:

• Opposing angles are congruent. The two angles across from each other have the same measure.

A rhombus has four interior angles, so their total is the same as any quadrilateral:

• The sum of the interior angles is 360°.
• Since opposing angles are equal, any two adjacent angles are supplementary, meaning they add to 180°.

Finally, rhombuses have a special diagonal property:

• The diagonals intersect perpendicularly. In other words, the diagonals cross at a right angle.

The following images of a rhombus show each of these properties:

Area of a rhombus

You can calculate the area of a rhombus in two main ways. The method you choose depends on what information you’re given.

Parallelogram method

This method uses the fact that a rhombus is a parallelogram. The area of a parallelogram is:

$$A=\text{base}\ast \text{height}$$

So, if you know the rhombus’s height (also called the altitude), you can use the same formula.

Be careful: you do not multiply the base by the side length. The side is slanted, so it is generally longer than the vertical height.

The area formula is:

$$A=\text{base}\ast \text{height}$$

Diagonal method

This method uses the lengths of the two diagonals. In a rhombus, the diagonals are not equal in length.

If you are given both diagonal lengths, the area is:

$$A=(\frac{1}{2})\ast {\text{diagonal}}_1\ast {\text{diagonal}}_2$$

Examples

Try the following examples by first choosing the best method and then finding the area of the rhombus.

Given the following rhombus with diagonal lengths of $5$ and $6$, what is the area?

What is the area of the following rhombus that has a side length of $5$ and a height of $4$?