How do you find the area of a parallelogram with vertices?
2 Answers
For parallelogram
Explanation:
Let's assume that our parallelogram
To determine the area of our parallelogram, we need the length of its base
First of all, to simplify the task, let's move it to a position when its vertex
So, we will perform the following transformation of coordinates:
Then the (
Our parallelogram now is defined by two vectors:
Determine the length of base
The length of altitude
The length
Angle
from which
Now we know all components to calculate the area:
Base
Altitude
The area is their product:
In terms of original coordinates, it looks like this:
another discussion
Explanation:
Geometric proof
Considering the figure
we can easily establish the formula for calculation of the area of a parallelogram ABCD, when any three vertices (say A,B,D) are known.
Since diagonal BD bisects the parallelogram into two congruent triangle.
The area of the parallelogram ABCD
= 2 area of triangle ABD
=2[ area of trapezium BAPQ +area of trap BQRD - area of trap DAPR]
=2[
=
=
=
This formula will give the area of the parallelogram .
Proof considering vector
It can also be established considering
Now
Position vector of point A w.r,t the origin O,
Position vector of point B w.r,t the origin O,
Position vector of point D w.r,t the origin O,
Now
Area of the Parallelogram ABCD
Again
Area =
=
=
=
=
Thus we have the same formula