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!

rhombus as baseball diamond

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?

Answer:

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?

Answer:

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.

Rhombus with sides congruent and labelled
$ \overline{AB} \cong \overline{BC} \\ \overline{BC} \cong \overline{CD} \\ \overline{CD} \cong \overline{DA} $

Angles

Diagonals

Diagonals are perpendicular.

Diagonals of Rhombus are Perpendicular

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

Area of Rhombus

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

Practice Problems

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?

Rhombus Diagonals

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

Rhombus Diagonals

The angles, are perpendicular!

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

Rhombus Diagonals

The diagonals of parallelograms bisect each other.

Therefore, ZA = 12, ZT = 5

Rhombus Diagonals

What kind of triangle is triangleZTA?

Rhombus Diagonals

A Right Triangle!

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

Rhombus Diagonals

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?

Rhombus Diagonals

Since, all 4 sides must be 13.

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

Rhombus Diagonals
Problem 1

If side WX = 22, what is WZ?

Rhombus Picture
WZ = 22
Problem 2

If side MN of rhombus LMNO is X + 5 and side LM is 2x − 9, what must be the value of x?

Sides of A Rhombus
Rhombus Sides Equation
Problem 3

What must be the value of x, if side BA = 5x-11
and side AD = 6x-18?

Properties of rhombus

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

Rhombus Diagrams
Problem 4

Is the four-sided shape below, MNOP, a rhombus? If not, classify the shape.

Rhombus and Parallelograms Comparison

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 5

What is the measure of the following angles in rhombus ABCD?

Angle Symbol ACD = 46° Angle Symbol ABD = 44°

Rhombus Angles Bisect
Problem 6

A generalization about the angles of a rhombus

You can think of a rhombus as four triangles that are created by the diagonals such as

Two Diagonals

What is true about the outside angles in each triangle? An examples of outside angles are

Two Angles

Rhombus Generalization

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

Problem 7

What is the value of x if Angle Symbol BCA = 3x -2 and Angle Symbol ACD = 12 + x ?

Rhombus

Since diagonals bisect vertex angles, angleBCA congruent angleACD

Equation
Problem 8

What is the value of x, given the angle measurements below?

Rhombus Angles Practice Problem
Equation
Problem 9

What is the area of HIJK?

Rhombus Area

Area = ½(IK × HJ) = ½ (9 × 12) = 54

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