*In the topic "mensuration", the foundation focuses on learning • what is measurement standard? • Absolute and Derived Standards • Measurement by superimposition • Measurement by calculation • Measurement by equivalence (infinitesimal and Cavalieri's principle in 2D and 3D. )*

The area of a triangle is derived by "Measurement by calculation". In this chapter, the different configurations of triangles are illustrated and a common formula is derived.

*click on the content to continue..*

Consider the triangle given in the figure. The base is `13cm` and height is `8cm`. Which of the following helps to find the area of the triangle?

- compare the triangle with a rectangle of `13cm` by `8cm` to calculate the area
- compare the triangle with a rectangle of `13cm` by `8cm` to calculate the area
- approximate the area using unit-squares

The answer is "compare the triangle with a rectangle of `13cm` by `8cm` to calculate the area".

The triangle of base `13cm` and height `8cm` is placed over a rectangle of length `13cm` and width `8cm`. It is noted that the triangle splits the rectangle into equal halves. What is the area of the triangle?

- half of the area of rectangle
- `1/2 xx text( base ) xx ( height)`
- both the above
- both the above

The answer is "both the above"

Consider the triangle given in the figure. The base is `15cm` and height is `6cm`. Which of the following helps to find the area of the triangle?

- compare the triangle with a rectangle of `15cm` by `6cm` to calculate the area
- compare the triangle with a rectangle of `15cm` by `6cm` to calculate the area
- approximate the area using unit-squares

The answer is "compare the triangle with a rectangle of `15cm` by `6cm` to calculate the area"

The triangle of base `15cm` and height `6cm` is placed over a rectangle of length `15cm` and width `6cm`.

The rectangle is visualized into two parts `ARCP` and `RBQC`

The triangle is visualized into two parts `/_\ARC` and `/_\RBC`

It is noted that the area of the triangle `/_\ARC` is half of the rectangle `ARCP` and area of the triangle `/_\RBC` is half of the rectangle `RBQC`. What is the area of the triangle?

- half of the area of rectangle
- `1/2 xx text( base ) xx ( height)`
- both the above
- both the above

The answer is "both the above".

The area of the triangle `/_\ABC` is sum of area of the two triangles `/_\ARC` and `/_\CRB`.

`=` half the area of rectangles `ARCP` and `CRBQ`.

`= 1/2 xx bar(AR) xx bar(RC) + 1/2 xx bar(RB) xx bar(RC)`

`= 1/2 xx bar(RC) xx ( bar(AR)+ bar(RB)) `

`= 1/2 xx bar(RC) xx bar(AB))`

`= 1/2 xx text( base )xx text( height)`

Consider the triangle given in the figure. The base is `8cm` and height is `6cm`. Which of the following helps to find the area of the triangle?

- compare the triangle with a rectangle covering `AQC` to calculate the area
- compare the triangle with a rectangle covering `AQC` to calculate the area
- approximate the area using unit-squares

The answer is "compare the triangle with a rectangle covering `AQC` to calculate the area"

The triangle of base `8cm` and height `6cm` is placed over a rectangle `AQCP`. The rectangle is visualized to two triangles `/_\AQC` and `/_APC`. The area of triangle `/_\AQC` is half of the rectangle. What is the area of the triangle `/_\ABC`?

- only a part of the area `/_\AQC`
- `1/2 xx text( base ) xx ( height)`
- both the above
- both the above

The answer is "both the above".

The area of the triangle `/_\ABC` is

`=` area of `/_\AQC - ` area of `/_\BQC`

`= 1/2 xx bar(AQ) xx bar(QC) - ` ` 1/2 xx bar(BQ) xx bar(QC)`

`= 1/2 xx bar(QC) xx ( bar(AQ) -bar(BQ) )`

`= 1/2 xx bar(QC) xx bar(AB)`

`= 1/2 xx text( base )xx text( height)`

Consider the three types of triangles,

For all the configurations, the area is derived as

`1/2 xx text( base ) xx ( height)`

**Area of a Triangle**:

`1/2 xx text( base ) xx ( height)`

*Solved Exercise Problem: *

What is the area of the triangle with base `2`cm and height `3`cm?

- `2xx3=6cm^2`
- `1/2 xx 2xx3 = 3cm^2`
- `1/2 xx 2xx3 = 3cm^2`

The answer is "`1/2 xx 2xx3 = 3cm^2`"

*slide-show version coming soon*