Find the area of a rhombus with a perimeter of 20 meters and a diagonal of 8 meters
To find the area of a rhombus, you need to know the lengths of its diagonals. However, in this case, we are given the perimeter and one diagonal. Therefore, we need to find the other diagonal before calculating the area.
Given information:
Perimeter of the rhombus = 20 meters
Diagonal = 8 meters
Step 1: Finding the length of each side
To find the length of each side, divide the perimeter by 4 (since a rhombus has four equal sides):
Length of each side = Perimeter / 4 = 20 m / 4 = 5 meters
Step 2: Finding the length of the other diagonal using the Pythagorean theorem
Since a rhombus is symmetric, each diagonal divides the rhombus into two congruent right triangles. We can use the Pythagorean theorem to find the length of the other diagonal:
The diagonal, perimeter, and side of the rhombus form a right triangles as shown:
(diagonal / 2) -------- side --------
| |
| |
| diagonal |
Using the Pythagorean theorem, we have:
(side/2) ^ 2 + (diagonal/2) ^ 2 = (diagonal) ^ 2
(5/2) ^ 2 + (diagonal/2) ^ 2 = 8 ^ 2
(25/4) + (diagonal/2) ^ 2 = 64
(diagonal/2) ^ 2 = 64 - 25/4
(diagonal/2) ^ 2 = 256/4 - 25/4
(diagonal/2) ^ 2 = 231/4
Taking the square root on both sides, we get:
diagonal/2 = sqrt(231/4)
diagonal/2 = sqrt(231) / 2
diagonal = 2 * sqrt(231) / 2
diagonal = sqrt(231)
Now we have both diagonals of the rhombus, with one being 8 meters and the other being sqrt(231) meters.
Step 3: Calculating the area of the rhombus
The area of a rhombus is given by the product of its diagonals divided by 2:
Area = (diagonal1 * diagonal2) / 2
Area = (8 m) * (sqrt(231) m) / 2
Simplifying, we have:
Area = 4 * sqrt(231) square meters
So, the area of the rhombus is 4 * sqrt(231) square meters.