# Rhombus: Properties and Shape

## Sides, Angles and Diagonals

Probably the most famous rhombus out there is the baseball diamond. The distance between each base is the same, making the shape a rhombus!

More interesting math facts!

A rhombus is a type of parallelogram, and what distinguishes its shape is that all four of its sides are congruent.

There are several formulas for the rhombus that have to do with its:

• Sides (click for more detail)

All 4 sides are congruent.

• Angles

Diagonals bisect vertex angles.

• Diagonals

Diagonals are perpendicular.

• Area

#### Is a Square a Rhombus?

Yes, a square is a rhombus

A square must have 4 congruent sides. Every rhombus has 4 congruent sides so every single square is also a rhombus. A square is a special rhombus that also has 4 right angles.

Keep in mind that the question "Is a square a rhombus?" means Is every square also always a rhombus?

#### Is Rhombus a Square?

No, a rhombus is not a square

A square must have 4 right angles. A rhombus, on the other hand, does not have any rules about its angles, so there are many many, examples of a rhombus that are not also squares.

Keep in mind that the question "Is a rhombus a square?" means Is every rhombus also always a square?

Diagram 2
As you can see in diagram 2, it is possible to create a rhombus that is not a square .

### Sides

All sides of a Rhombus are congruent.

$\overline{AB} \cong \overline{BC} \\ \overline{BC} \cong \overline{CD} \\ \overline{CD} \cong \overline{DA}$

### Angles

The diagonals bisect the vertex angles of a rhombus.

### Diagonals

Diagonals are perpendicular.

$$\angle AOD = 90^{\circ} \\ \angle AOB = 90^{\circ} \\ \angle BOC = 90^{\circ} \\ \angle COD = 90^{\circ} \\$$

### Area of Rhombus

Alex please put the rhombus calculator here input 1) side length 2) output -- angle measurements and area

### Putting It All Together

STAR is a rhombus. The measure of diagonals SA is 24 and the measure of TR is 10, what is the perimeter of this rhombus?

Ask yourself: What is true about the angles formed by the diagonals of a rhombus?

The angles, are perpendicular!

Now, what is true about the diagonals of all parallelograms?

The diagonals of parallelograms bisect each other.

Therefore, ZA = 12, ZT = 5

What kind of triangle is ZTA?

A Right Triangle!

So you can use the Pythagorean theorem to find the measure of side TA

Now, that you know the length of TA? How can you use the fact that the sides of a rhombus are congruent to finish this problem?

Since, all 4 sides must be 13.

The perimeter = 13 + 13 + 13 +13 = 52

WZ = 22
##### Problem 3

Since this shape is a rhombus you can set any of its sides equal to each other.

##### Problem 4

The shape below is not a rhombus because its diagonals are not perpendicular.

However, since opposite sides are congruent and parallel, and the diagonals bisect each other. The shape below is a parallelogram.

##### Problem 6

Since the diagonals of a rhombus are perpendicular, these outside angles must be complementary angles.

##### Problem 7

Since diagonals bisect vertex angles, BCA ACD

##### Problem 9

Area = ½(IK × HJ) = ½ (9 × 12) = 54