area of the 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 form right angles. Let's see what such a rhombus looks like:

We can see that it's kind of a square, kind of flattened.

The formula for the area of a rhombus

The area of a rhombus whose diagonals are p and q is equal to:

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

Calculator: calculate the area of the rhombus

How did we figure this out?

We can see that the diagonals p and q (the dashed lines) divide our rhombus into four triangles:

These triangles are identical, they are just rotated differently. So we can take the top two triangles, rotate them and move them down. We get a rectangle:

This rectangle has the same content as our rhombus - it's made up of the same four triangles that we just rearranged a little bit. We just need to calculate the area of this rectangle and we have the area of the rhombus. Yet we know that we calculate the area of a rectangle as the product of its shorter and longer sides. In our case, the area of the rectangle is equal to

$$S_\square=\Large |AC| \cdot |AB_1|$$

The length of the side AC is equal to the length of the horizontal diagonal p from the original figure. The length of the line segment AB₁ is the same as the length of the line segment SD, and the length of the line segment SD is equal to half the length of the vertical diagonal p - because the diagonals bisect each other. Add to the formula:

$$S_\square=\Large q \cdot \frac{p}{2}=\frac{p\cdot q}{2}$$

So if our rhombus has diagonal lengths equal to p = 7 and q = 4, then the area of such a rhombus is equal to

$$S=\frac{7\cdot4}{2}=\frac{28}{2}=14$$

Since a rhombus is also a parallelogram, you can also calculate the area of a rhombus using the formula for calculating the area of a parallelogram, but you need to know the side length and height of the rhombus.