Area of Rhombus

If you saw a kite flying in the sky on a windy day, you'd probably say that the kite is shaped like a diamond. In Geometry, this diamond shape is called a rhombus: a geometric shape with four equal sides. A rhombus is both a parallelogram and a quadrilateral. More specifically, a rhombus is a four-sided plane figure (a quadrilateral), with opposite sides that are parallel to one another (a parallelogram). Its special characteristic of having four equal sides, however, is what sets rhombuses apart from other quadrilaterals.

Area of rhombus formula

We have a specialized formula to find the area of a rhombus. Consider the following rhombus with diagonals of length d₁ and d₂.

Rhombus with diagonals d1 and d2 - Nilabhro Datta, StudySmarter Originals

The area of the rhombus is given by the formula:

$Area\mathit{}\mathit{=}\mathit{}\frac{\mathit{1}}{\mathit{2}}{d}_{\mathit{1}}{d}_{\mathit{2}}$

Since a rhombus is a special case of a parallelogram, the formula for finding the area of a parallelogram also applies to a rhombus and can be used to find the area of any rhombus as well. Consider the following parallelogram.

Parallelogram with base b and height h - Nilabhro Datta, StudySmarter Originals

The area of a parallelogram is given by the formula:

Area of a parallelogram = b × h

where b = base, h = height

Now the value, b, is the length of the side AB, which is considered the base here. Conventionally, one of the longer sides of the parallelogram is taken to be the base. However, since all sides of a rhombus are equal, any side can be considered the base. Also the height or altitude would be the same no matter which side is taken as the base. In the parallelogram shown above, we can see that the sides are not all equal lengths, which means this parallelogram is not a rhombus.

Note that for area we always use square units. So if we are using the SI units for example, the SI unit of length is meter (m), which means that the SI unit of area would be square meter (m²).

Examples: Area of rhombus equation

Let us look at some example problems related to the area of rhombuses.

Special rhombus case: Area of a square

You may have noticed that a square also meets the definition of a rhombus (a quadrilateral with 4 equal sides). A square is in fact a special case of a rhombus because all 4 sides of a square are equal. In addition, all 4 angles of a square are right angles.

A square - a special case of a rhombus and a rectangle - Nilabhro Datta, StudySmarter Originals

As a rhombus, the area formula for rhombuses can be applied to squares. However, using the area formula for parallelograms is also an applicable method, since squares and rhombuses are both parallelograms too. Look at the above square and recall the formula for the area of a parallelogram:

Area = base × height

Let us consider the base as the side AB. Since the angles in a square are right angles, the height is considered to be the length of either of the sides, AD or BC. This means that the side of the base is the same as the size of the height, since all 4 sides of a square are equal. Thus, in the case of a square, the above formula can be reduced to:

Area = side × side