Area of a Rhombus — Using Diagonals & Base Height

A rhombus is a quadrilateral with all four sides equal, and it is also a type of parallelogram, so opposite sides are parallel. Because of that, two procedures find its area, and which one you follow depends entirely on the measurements you are given:

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.

When to use each method

Since every rhombus is a parallelogram, it follows the same base-height idea, $A = bh$, where $h$ is the perpendicular distance from the base to the opposite side, not a slanted side. The diagonal method exists because a rhombus has a special property: its diagonals are perpendicular and split the shape into four right triangles, giving

This formula uses the full diagonals, not the half-diagonals from the center to a vertex. Choose the diagonal route when diagonals are easier to measure, and the base-height route when a base and perpendicular height are given directly.

Step by step (diagonal method)

1. Read off both full diagonals $d_1$ and $d_2$.
2. Multiply them and divide by $2$.
3. Write the answer in square units.

A full worked example

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

Use the diagonal formula:

Substitute:

So the area is

To verify, split the rhombus into its four right triangles. Half of the diagonals are $5$ cm and $4$ cm, so each triangle has area $\frac{1}{2}(5)(4) = 10$, and four of them give $4 \cdot 10 = 40\ \text{cm}^2$, matching the first method.

If instead you are given a base of $7$ cm and a perpendicular height of $6$ cm, switch methods and use $A = bh = 7 \cdot 6 = 42\ \text{cm}^2$.

Where students get stuck, and how to check each step

• Half-diagonals vs. full diagonals (step 1): If a diagram shows half-diagonals from the center, double them before substituting into $A = \frac{d_1 d_2}{2}$.
• Side as height (base-height route): In a slanted rhombus, the side length and the perpendicular height differ. Do not replace $h$ with the slanted side unless the rhombus is actually a square or the side is marked perpendicular.
• Side times side: For a general rhombus, $s^2$ is not the area; that only works when the rhombus is a square.
• Units: Area is measured in square units, not plain units.

To make both procedures stick, solve one version with diagonals $12$ cm and $9$ cm, then a second with base $7$ cm and perpendicular height $6$ cm, and compare which measurements each route needed.