# CLASS-9AREA OF AN EQUILATERAL TRIANGLE

Area of an Equilateral Triangle

Let ABC be an equilateral triangle with side a and AD be the perpendicular from A on BC. Then D is the mid-point of BC i.e. BD = a/2 The right angled ABD, by Pythagoras theorem, we have –

As per the above condition, AB = a, and BD = a/2

So,       AD² = AB² - BD²

a                   a²

AD² =  a² - (-----)²  =  a² - ------

2                   4

4a² - a²          3a²

4                4

3a²         √3a

2²            2

√3a

2

1

So, the area of ABC = ------- X BC X AD

2

1               √3a

= ------- X a X --------

2                 2

√3a²

= -------

4

So, Area of Equilateral Triangle with side a units = √3/4 a² sq. units

Height of the Equilateral Triangle is = √3a/2 units

Example.1) Calculate the area of an equilateral triangle of side 20 cm, correct to two decimal places. Also find its height correct to one decimal places (take √3 = 1.732).

√3

Ans.)  Area of the triangle = (------- X a²) sq. units

4

√3

= (------- X 20 X 20) cm²

4

=   (√3 X 100) cm²

=   (1.732 X 100) cm² = 173.20 cm²

√3

Height of the triangle =  (------ X a) cm

2

√3                √3

So, ------ X a =  (------- X 20)

2                 2

= (√3 X 10) cm = (1.732 X 10) cm

= 17.32 cm = 17.3 cm     (Ans.)

Example.2)  Calculate the area of an equilateral triangle whose height is 14 cm. (take √3 = 1.73)

Ans.)  Let the side of the triangle be a cm.

√3

Then its height => (------ X a) cm =  14

2

28

So,   a  =  ------- cm ………………….(i)

√3

√3

Area of the triangle = (------- X a²) cm²

4

√3           28          28           7 X 28

= ------- X ------- X -------- = -----------

4           √3          √3             √3

7 X 28

= ---------- =  113.29 cm²

1.73

Hence the area of the equilateral triangle is 113.29 cm²      (Ans.)