The sides of a rhombus with angle of 60° are 6 inches. Find the area of the rhombus.
To find the area of a rhombus, you can use the formula:
Area = (diagonal1 * diagonal2) / 2
In this case, since the rhombus has angles of 60°, the diagonals are equal in length and form two congruent equilateral triangles within the rhombus.
Given that the sides of the rhombus are 6 inches, we can calculate the length of the diagonals by using the sine formula:
Diagonal = (side length) / sin(angle)
In this case, since the angle is 60°, we can calculate the length of the diagonal as follows:
Diagonal = 6 / sin(60°)
By using the sine of 60°, which is √3 / 2, we have:
Diagonal = 6 / (√3 / 2) = (6 * 2) / √3 = (12 / √3) inches
Since both diagonals of the rhombus are congruent, we can say that the length of both diagonals is (12 / √3) inches.
Now, we can calculate the area of the rhombus using the formula:
Area = (diagonal1 * diagonal2) / 2
Area = [(12 / √3) * (12 / √3)] / 2
Area = (144 / (√3)^2) / 2
Area = (144 / 3) / 2
Area = 48 / 2
Area = 24 square inches
Therefore, the area of the rhombus is 24 square inches.
To find the area of a rhombus, we can use the formula:
Area = (diagonal1 * diagonal2) / 2
However, since we are only given the lengths of the sides of the rhombus, we need to use some additional geometric properties.
In a rhombus, the diagonals bisect each other and create four right-angled triangles.
Using these right-angled triangles, we can find the lengths of the diagonals:
1. Since the rhombus has an angle of 60°, we know that the opposite side is equal in length to one of its diagonals.
Therefore, the length of one of the diagonals is 6 inches.
2. To find the length of the other diagonal, we can use the Pythagorean theorem.
We know that the sides of the right-angled triangle (formed by half of the other diagonal, half of one side of the rhombus, and the bisected angle) have the following lengths:
- One-half of the rhombus side = 6/2 = 3 inches
- The opposite side of the right angle = 3 inches
- The hypotenuse (other half of the diagonal) = ?
Using the Pythagorean theorem, we can calculate the hypotenuse length:
(3 inches)^2 + (3 inches)^2 = hypotenuse^2
9 inches^2 + 9 inches^2 = hypotenuse^2
18 inches^2 = hypotenuse^2
hypotenuse = √(18 inches^2) = 3√2 inches
Now that we have the lengths of the diagonals, we can calculate the area of the rhombus using the formula mentioned earlier:
Area = (diagonal1 * diagonal2) / 2
Area = (6 inches * 3√2 inches) / 2
Area = 18√2 square inches
Therefore, the area of the rhombus is 18√2 square inches.