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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:

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 $20{\mathrm{cm}}^{2}$ .

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:

$s+s+s+s=P$

$4s=P$

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 $m\angle \mathrm{DAB}$ :

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

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

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

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

b. If a rhombus has an area of 25 cm^{2} and a base
measuring 5 cm, what is its height?

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

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

$\mathrm{height}=25\phantom{\rule{2pt}{0ex}}{\mathrm{cm}}^{2}\xf75\phantom{\rule{2pt}{0ex}}\mathrm{cm}=5\phantom{\rule{2pt}{0ex}}\mathrm{cm}$

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

$\mathrm{Perimeter}=4\times \mathrm{side\; length}$

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

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:

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

Polygon Exterior Angle Sum Theorem

Common Core: High School - Geometry Flashcards

Common Core: High School - Geometry Diagnostic Tests

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Working with shapes like the rhombus requires students to apply multiple areas of mathematics including formulas, geometric properties, and algebraic problem-solving skills simultaneously. It's a great way for students to gain experience applying math in practical situations, but it can also feel overwhelming with so many things going on at once. If your student could use a helping hand, private tutoring is among the most powerful educational tools available. Contact the Educational Directors at Varsity Tutors today for more information on how tutoring could help the student in your life.

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