Area Calculation of Parallelogram with Perpendicular Diagonal
In this article, we discuss the method to calculate the area of a parallelogram when a diagonal is given to be perpendicular to one of its sides. Understanding this concept can be particularly useful in various mathematical and real-life problems. We will explore multiple methods to find the area of the parallelogram ABCD, given that its base AB is 30 cm and its diagonal BD is 24 cm, and BD is perpendicular to AD.
Method 1: Using the Formula for Area of a Parallelogram
The area of a parallelogram can be calculated using the formula:
Area base times; height
Here, the base AB 30 cm, and the diagonal BD 24 cm, which is perpendicular to AD. Since BD is perpendicular to AD, it serves as the height of the parallelogram.
Using the formula, we calculate the area as follows:
Area AB times; BD 30 cm times; 24 cm
Area 720 cm2
Method 2: Using Trigonometric Relations
We can also use trigonometry to find the area. Since angle ADB is a right angle, we can use the sine function to find the height of the parallelogram.
sin(A) opposite/hypotenuse BD/AB 24/30
angle A sin-1(24/30) 53.13°
AD √(AB2 - BD2) √(302 - 242) √(900 - 576) √324 18 cm
The height of the parallelogram (AD) is then:
Height AD times; sin(53.13°) 18 times; 0.8 14.4 cm
The area of the parallelogram is then calculated as:
Area AB times; height 30 cm times; 14.4 cm 432 cm2
Alternative Approach Using Right Triangle Properties
We can also use the Pythagorean theorem to find the length of AD. Since ADB is a right angle, we can use the Pythagorean theorem:
AD2 AB2 - BD2 302 - 242 546
AD √546 ≈ 18 cm
The area of the parallelogram is:
Area 2 times; area of triangle ABD 2 times; (BD times; AD / 2) 2 times; (24 times; 18 / 2) 432 cm2
Conclusion
We have seen that there are multiple ways to calculate the area of a parallelogram when a diagonal is perpendicular to one of its sides. In this specific case, we found that the area of parallelogram ABCD is 432 cm2. Understanding these methods can help in solving various geometric problems with ease.
Frequently Asked Questions (FAQ)
Q: How can I find the area of a parallelogram when a diagonal is not perpendicular to a side?
A: If the diagonal is not perpendicular to a side, you will need to use different methods such as finding the height using trigonometric functions or the cross product method in vector geometry.
Q: Can the same method be used if the diagonal is not perpendicular to any side?
A: No, the methods discussed in this article specifically use the property of a perpendicular diagonal. For other cases, the area can be found using the lengths of the sides and the included angle or other geometric properties.
Q: What if the parallelogram is not a right-angled parallelogram?
A: In a general parallelogram, the area can be found using the formula: Area base times; height. The height can be found using trigonometric relations or the length of the sides and the included angle.