Area of a rhombus

A rhombus is similar to a square. It is a quadrilateral whose four sides are all the same length, but unlike a square, the sides of a rhombus do not meet at right angles. Let's see what a rhombus looks like:

We can see that it is a square that has been squashed a little.

Formula for the area of a rhombus

The area S of a rhombus whose diagonals have lengths p and q is

$$S=\frac{p\cdot q}{2}$$

Calculator: work out the area of a rhombus

How did we get this?

Notice that the diagonals p and q (the dashed line segments) split the rhombus into four triangles:

These triangles are congruent; they are just turned round differently. So we can take the two upper triangles, turn them and move them to the bottom. This gives us a rectangle:

This rectangle has the same area as our rhombus – it is made of the same four triangles, which we have just rearranged a little. So all we need is the area of this rectangle, and we have the area of the rhombus. We know that the area of a rectangle is the product of its shorter and longer sides. In our case, the area of the rectangle is

$$S_{◻}=|AC|\cdot |A{B}_1|$$

The length of side AC is equal to the length of the horizontal diagonal p in the original picture. The length of line segment AB₁ is the same as the length of line segment OD, and the length of OD is half the length of the vertical diagonal q, because the diagonals bisect each other. Substituting into the formula:

$$S_{◻}=p\cdot \frac{q}{2}=\frac{p\cdot q}{2}$$

So if our rhombus has diagonals p = 7 and q = 4, its area is

$$S=\frac{7\cdot 4}{2}=\frac{28}{2}=14$$

A rhombus is also a parallelogram, so you can find its area using the formula for the area of a parallelogram, but then you need to know the side length and the height of the rhombus.