Area of a Parallelogram
The area of a parallelogram is calculated using a simple formula: $$ A = b \times h $$ where \( A \) is the area, \( b \) is the length of the base, and \( h \) is the height, that is, the perpendicular distance between the base and the opposite side.
Although the formula is straightforward, it is based on an important geometric idea that has been known since ancient times. Greek mathematicians, including Euclid, studied parallelograms extensively and established many of the properties that are still taught in geometry today.
A Practical Example
Suppose a parallelogram has a base of 5 meters and a height of 3 meters.
To find its area, simply multiply the base by the height:
$$ A = 5 \, m \times 3 \, m = 15 \, m^2 $$
Therefore, the parallelogram covers an area of 15 square meters.
Why Does the Formula Work?
The reason behind the formula becomes clear if we transform the parallelogram into a rectangle.
Start by drawing the altitude from vertex D to the base.
Next, cut along triangle ADH and move that triangular section to the opposite side of the figure.
This rearrangement produces rectangle HDCE.
The new rectangle has exactly the same base and height as the original parallelogram. Since no part of the figure has been added or removed, both shapes have the same area.
Because the area of a rectangle is equal to its base multiplied by its height, the area of a parallelogram is also:
$$ A = b \times h $$
Finding the Area from Cartesian Coordinates
When the coordinates of the vertices are known, the area of a parallelogram can be calculated without measuring its base or height directly.
If the vertices are \( (x_1, y_1) \), \( (x_2, y_2) \), \( (x_3, y_3) \), and \( (x_4, y_4) \), listed in cyclic order, the area can be found using the shoelace formula:
$$ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| $$
This method is particularly useful in coordinate geometry and computer graphics, where the positions of points are often known precisely.
Example
Consider a parallelogram whose vertices are:
$$ A(1, 2) $$
$$ B(4, 5) $$
$$ C(7, 4) $$
$$ D(4, 1) $$
Writing the coordinates in cyclic order gives:
$$ (x_1, y_1) = (1, 2) \\
(x_2, y_2) = (4, 5) \\
(x_3, y_3) = (7, 4) \\
(x_4, y_4) = (4, 1) $$
Applying the formula:
$$ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| $$
Substituting the coordinates:
$$ A = \frac{1}{2} \left| 1 \cdot 5 + 4 \cdot 4 + 7 \cdot 1 + 4 \cdot 2 - (2 \cdot 4 + 5 \cdot 7 + 4 \cdot 4 + 1 \cdot 1) \right| $$
Simplifying the expression:
$$ A = \frac{1}{2} \left| 5 + 16 + 7 + 8 - (8 + 35 + 16 + 1) \right| $$
$$ A = \frac{1}{2} \left| 36 - 60 \right| $$
$$ A = \frac{1}{2} \left| -24 \right| $$
$$ A = \frac{1}{2} \cdot 24 $$
$$ A = 12 $$
The area of the parallelogram is therefore 12 square units.
We can verify this result using the standard formula based on the base and height:
$$ A = b \times h = 4.2426 \times 2.8284 = 12 $$
Both methods produce the same result, confirming that the area of the parallelogram is 12 square units.
