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# Rhombus

As you continue to study geometry, you'll start to encounter funny-looking shapes that you may not have worked with before. For example, the rhombus is a parallelogram with four congruent sides. Here are a couple of examples:

If you're looking at multiple of these shapes, the accepted plural form of a rhombus is rhombi. Rhombi might look strange, but they have a lot of interesting properties to explore. Another fact that is going to be of help to us is that the opposite angles of a rhombus are congruent with each other.

Let's take a closer look at some math involving the rhombus:

## Calculating the area of a rhombus

As with other geometric shapes, one of the first things you might do with a rhombus is to calculate its area. The formula for calculating the area of a rhombus is the same as any other parallelogram:

A=bh, where b is the length of a base and h, is the height. Consider the following diagram:

Each base measures 5 cm and the height is 4 cm, so we multiply the two figures to get an area of $20{\mathrm{cm}}^{2}$ .

## Calculating the perimeter of a rhombus

Similarly, the perimeter of a rhombus uses the same formula as any other parallelogram: adding all four sides together. If one side measures 3 inches, that means all four sides will measure 3 inches and we can get the perimeter by multiplying 3 by 4. This can be expressed in two different ways mathematically:

$s+s+s+s=P$

$4s=P$

## Working with the diagonals of a rhombus

The diagonals of a rhombus are always perpendicular. Here is an illustration:

In this example, we could say that if PQRS is a rhombus, then the diagonal line PR is perpendicular to the diagonal line QS. We can express this mathematically using a ⊥ symbol between the two lines.

Furthermore, if a parallelogram is a rhombus then each diagonal bisects a pair of opposite angles. Returning to the example above, if PQRS is a rhombus then $\angle 1\cong \angle 2,\phantom{\rule{2pt}{0ex}}\angle 3\cong \angle 4,\phantom{\rule{2pt}{0ex}}\angle 5\cong \angle 6$ , and $\angle 7\cong \angle 8$ .

Finally, consecutive interior angles of a rhombus are always supplementary.

We can use this information to figure out the measure of other angles. Consider the following diagram:

If $m\angle \mathrm{ABC}=2x-7$ and $m\angle \mathrm{BCD}=2x+3$ , we know that the two equations added together must equal 180 degrees:

$2x-7+2x+3=180$

Next, we can simplify the algebraic expression into:

$4x-4=180$

$4x=184$

From here, a little division will tell us that $x=46$ . Now that we know this and that the opposite angles of a rhombus are congruent, we can figure out the measure of $m\angle \mathrm{DAB}$ :

$m\angle \mathrm{DAB}=m\angle \mathrm{BCD}=2\left(46\right)+95\phantom{\rule{2pt}{0ex}}\text{degrees}$

## Rhombus practice questions

a. If the base of a rhombus measures 8 inches and its height is 4 inches, what is its area?

$\mathrm{Area}=\mathrm{base}×\mathrm{height}$

Area = $8\phantom{\rule{2pt}{0ex}}\mathrm{inches}×4\phantom{\rule{2pt}{0ex}}\mathrm{inches}=32\phantom{\rule{2pt}{0ex}}{\mathrm{inches}}^{2}$

b. If a rhombus has an area of 25 cm2 and a base measuring 5 cm, what is its height?

$\mathrm{Area}=\mathrm{base}×\mathrm{height}$

$25\phantom{\rule{2pt}{0ex}}{\mathrm{cm}}^{2}=5\phantom{\rule{2pt}{0ex}}\mathrm{cm}×\mathrm{height}$

$\mathrm{height}=25\phantom{\rule{2pt}{0ex}}{\mathrm{cm}}^{2}÷5\phantom{\rule{2pt}{0ex}}\mathrm{cm}=5\phantom{\rule{2pt}{0ex}}\mathrm{cm}$

c. What is the perimeter of a rhombus with one side measuring 6 miles?

$\mathrm{Perimeter}=4×\mathrm{side length}$

$\mathrm{Perimeter}=4×6\phantom{\rule{2pt}{0ex}}\mathrm{miles}=24\phantom{\rule{2pt}{0ex}}\mathrm{miles}$

d. What useful properties do the diagonals of a rhombus have?

The diagonals are perpendicular and bisect each other

e. If you add the measurements of two consecutive interior angles of a rhombus, what do you get?

The sum of two consecutive interior angles of a rhombus equals 180 degrees

f. Considering the following diagram, find $m\angle \mathrm{YZV}$ assuming XYZW is a rhombus.

Since the diagonals are perpendicular $m\angle \mathrm{YZV}=90$

the sum of the angles of a triangle is 180, so:

$180-90-63=27\phantom{\rule{2pt}{0ex}}\mathrm{degree}$

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