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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 Parallelograma1

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. DL is the altitude corresponding to the base AB of parallelogram ABCD ; and
  2. DM is the altitude corresponding to the base BC of the parallelogram ABCD .

 m5x

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.a2

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, ABCD \ and\  ABEF are two parallelograms. They have the same base AB and are between the same parallels AB \ and\  FC . This means that the altitude for both is the same.

\therefore (\parallel gm \ ABCD)=Area(\parallel gm \ ABEF)

Corollary:  Since a rectangle is also a parallelogram. Therefore we can use the above theorem to say that:a3

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 ABCD and a rectangle ABEF on the same base AB and between the same parallel lines AB \ and\  FC . Then,

Area(\parallel gm\ ABCD)=Area(\parallel gm\ ABEF)=AB\times AF (base \times height)

Hence,

Area \ of \ Parallelogram=base\times height

m6x

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

Area \ of \ the \ triangle=\frac{1}{2}(Area \ of \ the \ parallelogram)

In each of the adjoining figures, \Delta ABC \ and\ \parallel \ gm\ BCDE are on the same base BC and between the same parallels BC \ and\ AD .

\therefore (\Delta ABC)=\frac{1}{2}(Area \ of \ the \ \parallel \ gm \ BCDE) a5

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

Consider a \Delta ABC  and on the same base BC and between the same parallels BC \ and \ AD .

Then, using the above result, we have:

\therefore Area(\Delta ABC)=\frac{1}{2}\times Area (\parallel \ gm \ ABCD)

=\frac{1}{2} \times BC \times AL [Using the formula of the area of ||gm]

a6But, AL is the height corresponding to base BC of \Delta ABC .

Hence, Area\ of\ Triangle= \frac{1}{2}\times base \times height .

 

Theorem 3: Two triangles on the same base and between the same parallels are equal in area.a7

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

\therefore \ Area(\Delta ABC)=Area(\Delta DBC)

 

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

In the adjoining figurea8

\Delta ABC \ and\  \Delta DEF  have equal base BC=EF=3\ cm (or any measurement) and they are between the same parallels BF \ and \ AD .

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

\therefore Area (\Delta ABC)= Area (\Delta DEF)

m7x

 

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