How to find the area of a rhombus with one diagonal and perimeter?
The area of the rhombus is the product of the lengths of its diagonals.
Answer: We can make use of the unknown diagonal's formula √(P2/16 - d12/4) × 4, where P is perimeter and d1 is the known diagonal of the rhombus.
Let us go through the steps to understand the explanation.
What is the Area of a Rhombus?
The space enclosed by a rhombus in a two-dimensional space is called the area of a rhombus. A rhombus is a type of quadrilateral projected on a two-dimensional (2D) plane, having four sides that are equal in length and are congruent.
The area of the rhombus as stated above can be easily calculated using only the diagonals of the rhombus.
But, if we do not know the length of any one of the diagonals, we need to calculate that diagonal's length first.
Suppose, we are given the perimeter P of the rhombus and one of the diagonal d1.
To calculate diagonal d2
Since all 4 sides of the rhombus are the same, so the length of each side = P/4
Using the property of diagonals, which is that they intersect at right angles to each other, we can calculate the other diagonal d2.
Taking triangle AOB into consideration.
Applying Pythagoras theorem on it.
(AO)2 + (BO)2 = (AB)2
(d1/2)2 + (d2/2)2 = (Side)2
(d1/2)2 + (d2/2)2 = (P/4)2
(d2)2 = (P2/16 - d12/4) × 4
d2 = √(P2/16 - d12/4) × 4
Now applying the formula of the area of the rhombus, using two diagonals value, we can easily calculate its area.