Rhombus: Properties and Shape

Sides, Angles and Diagonals

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(click for more detail)
    • diagonals bisect vertex angles
  • Diagonals(click for more detail)
    • diagonals are perpendicular
  • Area (click for more detail)

A square is a rhombus and a rectangle. In other words, if each angle of a rhombus is 90°, then it's a square.

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

Rhombus Sides

All sides of a Rhombus are congruent.

AB congruent BC congruent CD congruent AD

Rhombus Practice Problems

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.

Diagonals of a Rhombus

Diagonals of Rhombus are Perpendicular
Diagonals are perpendicular.
  • AOD = 90°
  • AOB = 90°
  • BOC = 90°
  • COD = 90°
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.

Angles of a Rhombus

The diagonals bisect the vertex angles of a rhombus. A proof of this property of the diagonals

Rhombus Angles
Problem 5

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

ACD = 46°
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

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

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

Problem 7

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


Since diagonals bisect vertex angles, angleBCA angleACD

Problem 8

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

Rhombus Angles Practice Problem

Area of Rhombus

The formula for area of A Rhombus
Area = ½(diagonal1 × diagonal2)
Formula area of rhombus
Problem 9

What is the area of HIJK?

Rhombus Area

Area = ½(IK ×HJ) = ½ (9 ×12) = 54

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