Notes of Area of Parallelograms and Triangles

Introduction

Two figures are called congruent, if they have the same shape and the same size. 

The part of the plane enclosed by a simple closed figure is called a planar region corresponding to that figure. The magnitude or measure of that planar region is called its area

If two figures A and B are congruent, they must have equal areas. 

Two figures having equal areas need not be congruent. 

The area represents the amount of planar surface being covered by a closed geometric figure.

1 5

Areas of closed figures

If a planner region formed by a figure T is made up of two non-overlapping planner regions formed by figures P and Q, then ar(T) = ar(P) + ar(Q). 

2 4

Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices (or the vertex) opposite to the common base of each figure lie on a line parallel to the base. 

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

3 5

In the figure, parallelograms PQCD and ARCD lie on the same base CD and between same parallels CD and PR. So, ar(PQCD) = ar(ARCD). 

Figures on the Same Base and Between the Same Parallels

Two figures are said to be on the same base and between the same parallels if:

  1. They have a common side.
  2. The sides parallel to the common base and vertices opposite the common side lie on the same straight line parallel to the base.
4 5

Figures on same base AB and between same parallels AB and PQ

For example : Parallelogram  ABCD, Rectangle ABEF and Triangles ABP and ABQ

Area of a parallelogram is the product of its any side and the corresponding altitude. 

Parallelograms on equal bases and between the same parallels are equal in area. 

Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels. 

If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half of the area of the parallelogram. 

Two triangles on the same base (or equal base) and between the same parallel are equal in area. 

5 3

In the figure, triangles ABC and PBC lie on the same base BC and between same parallels BC and AP. So, ar(ABC) = ar(PBC). 

Area of triangle is half the product of its base (or any side) and the corresponding altitude (or height). 

Two triangles with same base (or equal bases) and equal areas will have equal corresponding altitudes. 

Two triangles having the same base (or equal bases) and equal areas lie between the same parallels. 

A median of a triangle divides it into triangles of equal areas.

Area of a parallelogram

6 2

Parallelogram

Area of a parallelogram = b×h

Where ‘b′ is the base and ‘h′ is the corresponding altitude(Height).

Area of a triangle

7 2

Area of triangle

Area of a triangle = 12×b×h

Where \(‘b’\) is the base and \(‘h’\) is the corresponding altitude.

Theorems

Parallelograms on the same Base and Between the same Parallels

Two parallelograms are said to be on the same base and between the same parallels if

  1. They have a common side.
  2. The sides parallel to the common side lie on the same straight line.
8 2

Parallelogram ABCD and ABEF

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

Here, ar(parallelogram ABCD)=ar(parallelogram ABEF)

Triangles on the same Base and between the same Parallels

Two triangles are said to be on the same base and between the same parallels if

  1. They have a common side.
  2. The vertices opposite the common side lie on a straight line parallel to the common side.
9 2

Triangles ABC and ABD

Theorem : Triangles that lie on the same base and between the same parallels are equal in area.

Here, ar(ΔABC)=ar(ΔABD)

Two triangles having the same base & equal areas

If two triangles have the same base and are equal in area, then, their corresponding altitudes are equal.

10 2

If two triangles have equal bases and are equal in area, then their corresponding altitudes are equal.

A Parallelogram and a triangle between the same parallels

A triangle and a parallelogarm are said to be on the same base and between the same parallels if

  1. They have a common side.
  2. The vertices opposite the common side lie on a straight line parallel to the common side.
11 2

A triangle ABC and a parallelogram ABDE

Theorem : If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half the area of the parallelogram.

Here ar(ΔABC)=12ar(parallelogarm ABDE)