How to find the area of a rhombus with one diagonal and perimeter?
The area of a rhombus is the product of the lengths of its diagonals.
Answer: When the perimeter and one diagonal of a rhombus is given, we can use the Pythagoras theorem to find the other diagonal and then calculate its area.
Let us go through the steps to understand the explanation.
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.
However, if we do not know the length of any one of the diagonals, we need to calculate that diagonal's length first.
Suppose, we know the perimeter P of the rhombus and one of the diagonals \(d_{1}\)
To calculate diagonal \(d_{2}\)
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 \(d_{2}\)
Taking triangle AOB into consideration.
Applying Pythagoras theorem to it.
(AO)2 + (BO)2 = (AB)2
( \(d_{1}\) /2)2 + ( \(d_{2}\) /2)2 = (Side)2
( \(d_{1}\) /2)2 + ( \(d_{2}\) /2)2 = (P/4)2
( \(d_{2}\) )2 = (P2/16 - \(d_{1}\) 2/4) × 4
\(d_{2}\) = √[(P2/16 - \(d_{1}\) 2/4) × 4]
Now applying the formula of the area of the rhombus, using the values of both the diagonals, we can easily calculate its area.