__Properties of a Rhombus__

These are some of the most important properties of a rhombus.

Consider the rhombus ABCD shown in the figure above.

1 - All sides are congruent (equal lengths).

length AB = length BC = length CD = length DA = a.

2 - Opposite sides are parallel.

AD is parallel to BC and AB is parallel to DC.

3 - The two diagonals are perpendicular.

AC is perpendicular to BD.

4 - Opposite internal angles are congruent (equal sizes).

internal angle A = internal angle C

and

internal angle B = internal angle D.

5 - Any two consecutive internal angles are supplementary : they add up to 180 degrees.

angle A + angle B = 180 degrees

angle B + angle C = 180 degrees

angle C + angle D = 180 degrees

angle D + angle A = 180 degrees

__Area of a Rhombus__

These are three formulas for the area of the rhombus.

__formula 1:__

**area = a*h** , where **a** is the side length of the rhombus and **h** is the perpendicular distance between two parallel sides of the rhombus.

__formula 2:__

**area = a**^{ 2}*sin (A) = a^{ 2}*sin (B). Since angles **A** and **B** are supplementary angles, sin (A) = sin (B).

__formula 3:__

**area = (1/2)*d1*d2**, where **d1** and **d2** are the lengths of the two diagonals.

We now present some problems with detailed solutions.

__Problem 1:__ The size of the obtuse angle of a rhombus is twice the size of its acute angle. The side length of the rhombus is equal to 10 feet. Find its area.

__Solution to Problem 1:__

- A rhombus has 2 congruent opposite acute angles and two congruent opposite obtuse angles. One of the properties of a rhombus is that any two internal consecutive angles are supplementary. Let x be the acute angle. The obtuse angle is twice: 2x. Which gives the following equation.

x + 2 x = 180 degrees.

- Solve the above equation for x.

3x = 180 degrees.

x = 60 degrees.

- We use the formula for the area of a triangle that uses the side lengths and any one of the angles then multiply the area by 2.

area of rhombus = 2 (1 / 2) (10 feet)^{ 2} sin (60 degrees)

= 86.6 feet^{ 2} (rounded to 1 decimal place)

__Problem 2:__ The lengths of the diagonals of a rhombus are 20 and 48 meters. Find the perimeter of the rhombus?

__Solution to Problem 2:__

- Below is shown a rhombus with the given diagonals. Consider the right triangle BOC and apply Pythagora's theorem as follows

BC^{ 2} = 10^{ 2} + 24^{ 2}

- and evaluate BC

BC = 26 meters.

- We now evaluate the perimeter P as follows:

P = 4 * 26 = 104 meters.

__Problem 3:__ The perimeter of a rhombus is 120 feet and one of its diagonal has a length of 40 feet. Find the area of the rhombus.

__Solution to Problem 3:__

- A perimeter of 120 when divided by 4 gives the side of the rhombus 30 feet. The length of the side OC of the right triangle is equal to half the diagonal: 20 feet. Let us now consider the right triangle BOC and apply Pythagora's theorem to find the length of side BO.

30^{ 2} = BO^{ 2} + 20^{ 2}

BO = 10 sqrt(5) feet

- We now calculate the area of the right triangle BOC and multiply it by 4 to obtain the area of the rhombus.

area = 4 ( 1/2) BO * OC = 4 (1/2)10 sqrt (5) * 20

= 400 sqrt(5) feet^{ 2}

More references on geometry.

Geometry Tutorials, Problems and Interactive Applets.

Rhombus - Geometry Calculator. Calculator to calculate the characteristics of a rhombus.