Area

The amount of surface enclosed by the sides of a closed bounded figure is called its area.

Definition of Base and Altitude of a Parallelogram

Any side of a parallelogram can be called its **base**.

The length of the line segment which is perpendicular to the Base from the opposite side is called the altitude or height of the parallelogram corresponding to the given base.

In the adjoining figure we see that,

1. is the altitude corresponding to the base of parallelogram ; and

2. is the altitude corresponding to the base of the parallelogram .

** **Area of a Parallelogram: If you see the figure below, then you could cut the left side of figure (blue triangle) and move it to the right side. It will make it a rectangle.

Hence you can calculate the area of the Parallelogram as. This is a pictorial way of coming up with a formula for area of Parallelogram.

Source: https://en.wikipedia.org/wiki/Parallelogram#Area_formula

There is a very good animation of the above diagram:

Source: https://en.wikipedia.org/wiki/Parallelogram#/media/File:Parallelogram_area_animated.gif

Theorem 1: Parallelograms on the same base and between the same parallels are equal in area.

In the adjoining figure, are two parallelograms. They have the same base and are between the same parallels . This means that the altitude for both is the same.

**Corollary: **Since a rectangle is also a parallelogram. Therefore we can use the above theorem to say that:

A parallelogram and a rectangle on the same base and between the same parallels are equal in area. *(Same base x height)*

With this we can derive the formula for the area of the triangle.

Consider a parallelogram and a rectangle on the same base and between the same parallel lines . Then,

Hence,

Theorem 2: If a triangle and a parallelogram are on the same base and between the same parallels, then

In each of the adjoining figures, are on the same base and between the same parallels .

We can even drive the area of a triangle using this theorem.

Consider a and on the same base BC and between the same parallels .

Then, using the above result, we have:

[Using the formula of the area of ||gm]

But, is the height corresponding to base of .

Hence, .

Theorem 3: Two triangles on the same base and between the same parallels are equal in area.

In the given figure, are on the same base and between the same parallels .

* *

Theorem 4: Two Triangles having equal base and lying between the same parallels are equal in area.

In the adjoining figure

have equal base (or any measurement) and they are between the same parallels .

This is because the base is the same and the height (distance between the parallel lines) is the same.