**Introduction of Areas of Parallelograms and Triangles Notes**

**Introduction**

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

Areas of closed figures

Figures on the Common Base and Between the Same Parallels

Two shapes are stated to be on the common base and between the same parallels if:

a) They have a **common side**.

b) The sides parallel to the common base and vertices opposite the common side **lie on the same straight line parallel to the base**.

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

Area of a parallelogram

**Parallelogram**

Area of a parallelogram = b×h

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

Area of a triangle

**Area of triangle**

Area of a triangle = (1/2)×b×h

Where **“b”** is the** base** and “**h”** is the corresponding **altitude**.

Theorems

Parallelograms on the Common Base and Between the Same Parallels

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

a) They have a **common side.**

b) The sides parallel to the common side **lie on the same straight line**.

Parallelogram ABCD and ABEF

**Theorem**: Parallelograms that lie on the **common base** and **between the same parallels** are said to have **equal in area**.

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

Triangles on the Common Base and Between the Same Parallels

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

a) They have a **common side**.

b) The vertices opposite the common side **lie on a straight line parallel to the common side**.

Triangles ABC and ABD

**Theorem**: Triangles that lie on the same or the common base and also between the same parallels are said to have an equal area.

Here, ar(ΔABC)=ar(ΔABD)

Two Triangles Having the Common Base & Equal Areas

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

a) They have a** common side.**

b) The vertices opposite the common side **lie on a straight line parallel to the common side**.

A triangle ABC and a parallelogram ABDE

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

Here, ar(ΔABC)=(1/2) ar(parallelogarm ABDE)