# How To Find The Height Of A Rhombus?

A rhombus is a quadrilateral whose four sides are all equal. Its diagonals are perpendicular bisectors, and its opposite angles are equal. The method used to calculate the height of a rhombus depends on the information we have about it. For instance, given the length of its side and its area, we can calculate its height using the formula (area)/(side).

To determine how to calculate the height of a rhombus, we must first define it.

A rhombus is a four-sided plane figure (quadrilateral) that has the following properties:

-All its sides are equal;

-Its opposite sides are parallel to each other;

-Its opposite angles are equal;

-Its diagonals bisect each other at right angles. These diagonals are also angle bisectors;

-Its diagonals divide it into 4 congruent (same shape and size) triangles.

The method used to determine the height of a rhombus depends on the dimensions given.

1) Given the side(s) of a rhombus and its area:

The formula for finding the area of a rhombus is side \times perpendicular height.

Thus, the height of the rhombus in this case is \frac{area}{side} .

Example: Find the height of a rhombus whose area is 20 cm^{2} and side is 4 cm.

Height = \frac{area}{side} = \frac{20}{4} = 5 cm.

2) Given the diagonals of a rhombus:

If the diagonals d_{1} and d_{2} of a rhombus are given, then using the following two properties of rhombi, namely, the diagonals bisect each other at right angles and the diagonals divide the rhombus into 4 congruent triangles, we can find the area of one triangle as follows:

Area of 1 triangle = \frac{1}{2} \times bh = \frac{1}{2} \times \frac{d_{1}}{2} \times \frac{d_{2}}{2} =\frac{d_{1}d_{2}}{8}

And area of the 4 triangles in the rhombus is \frac{d_{1}d_{2}}{8} \times 4 = \frac{d_{1}d_{2}}{2} .

Having found the area of the rhombus, we can next determine the side of the rhombus using Pythagoras Theorem.

The base of one of the 4 triangles formed by the diagonals is \frac{d_{1}}{2} and the height is \frac{d_{2}}{2} . Therefore, the hypotenuse, which forms the side of the rhombus, is \sqrt{(\frac{d_{1}}{2})^{2} +(\frac{d_{2}}{2})^{2}} .

Finally, we know that the area of the rhombus can also be calculated using the formula side x height. Here, we have already found the area and the side. Therefore, the height of the rhombus is \frac{area}{side} , where area = \frac{d_{1}d_{2}}{2} and side = \sqrt{(\frac{d_{1}}{2})^{2} +(\frac{d_{2}}{2})^{2}} .

Example: Find the height of a rhombus whose diagonals are 90 mm and 400 mm.

The area of the rhombus is \frac{d_{1}d_{2}}{2} = \frac{90 \times 400}{2} = 18000 mm^{2} .

One of the four congruent right-angled triangles in the rhombus has a height \frac{d_{1}}{2} = 45 mm and a base \frac{d_{2}}{2} = 200 mm. Therefore, the height of this triangle, which is also the side of the rhombus, is \sqrt{45^{2} + 200^{2}} = 205 mm.

Thus, using the formula area of rhombus = side x height, we obtain height = \frac{area}{side} = \frac{18000}{205} = 87.805 mm (to 3 decimal places).