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).

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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).

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