Area of a Rhombus — Using Diagonals & Base Height

The area of a rhombus is the space inside the shape. In most problems, you use one of these two formulas:

$$A = bh$$

and

$$A = \frac{d_1 d_2}{2}$$

Use $A = bh$ when you know a base and the perpendicular height. Use $A = \frac{d_1 d_2}{2}$ when you know the full diagonals.

Area of a rhombus formulas

A rhombus is a quadrilateral with all four sides equal. It is also a type of parallelogram, so opposite sides are parallel.

That matters because every rhombus follows the same base-height idea as a parallelogram:

$$A = bh$$

Here, $h$ is not a slanted side. It is the perpendicular distance from the base to the opposite side.

Why the diagonal formula works

If you know a base and the perpendicular height, the area comes straight from $A = bh$.

If you know the diagonals instead, a rhombus has a special property: its diagonals are perpendicular. They split the shape into four right triangles, so the total area becomes

$$A = \frac{d_1 d_2}{2}$$

This formula uses the full diagonals, not the half-diagonals from the center to a vertex.

Worked example using diagonals

Suppose a rhombus has diagonals $d_1 = 10$ cm and $d_2 = 8$ cm.

Use the diagonal formula:

$$A = \frac{d_1 d_2}{2}$$

Substitute the values:

$$A = \frac{10 \cdot 8}{2}$$ $$A = \frac{80}{2} = 40$$

So the area is

$$40\ \text{cm}^2$$

A quick check helps. Half of the diagonals are $5$ cm and $4$ cm, so each of the four right triangles has area

$$\frac{1}{2}(5)(4) = 10$$

Four such triangles give

$$4 \cdot 10 = 40\ \text{cm}^2$$

which matches the first method.

When to use base and height instead

Use $A = bh$ when the problem gives a base and a perpendicular height directly.

For example, if a rhombus has base $7$ cm and perpendicular height $6$ cm, then

$$A = bh = 7 \cdot 6 = 42\ \text{cm}^2$$

Do not replace the height with the slanted side unless the rhombus is actually a square or the side is marked perpendicular.

Common mistakes with rhombus area

Using a side length as the height

In a slanted rhombus, the side length and the perpendicular height are different. If you use $A = bh$, make sure $h$ is perpendicular to the base.

Forgetting that the diagonal formula needs full diagonals

If a diagram shows half-diagonals from the center, do not plug those values directly into $A = \frac{d_1 d_2}{2}$. Double them first to get the full diagonals.

Using side times side

For a general rhombus, $s^2$ is not the area. That works only in the special case where the rhombus is a square.

Dropping the square units

Area is measured in square units, not plain units.

When the area of a rhombus is used

The area of a rhombus appears in school geometry, coordinate geometry, tiling problems, and any diagram with a diamond-shaped region.

It is especially useful when diagonals are easier to measure than height. In other problems, base and perpendicular height are the cleaner route. The right formula depends on which measurements you actually have.

Try a similar problem

Try your own version with diagonals $12$ cm and $9$ cm. Then solve a second version with base $7$ cm and perpendicular height $6$ cm. Comparing those two setups helps the two formulas stick.