# Rhombus Problems

Rhombus problems with detailed solutions.

 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 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.