Definition of a Rhombus
The rhombus is a parallelogram with four congruent sides. A square is a special case of a rhombus.
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
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.
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.
area = a 2*sin (A) = a 2*sin (B). Since anglesA and B are supplementary angles, sin (A) = sin (B).
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.