Question

if each angle of an equileteral triangle is 60 degree and perimeter is 180 degree find its area by heron's formula?

Answer 1

Hola Mate!!

Your answer :-

P = a + a + a

S = p/2

S = a + a + a / 2

S = 3a/2

Area =
_________________________
/ 3a/2(3a/2 - a)(3a/2 - a)(3a/2 - a)
____________________
/ 3a/2 ( a/2 ) ( a/2 ) ( a/2 )
..................___
a/2 × a/2 / 3
............___
a^2/4 /3
__
/3 a^2 / 4

______________
$given = > \\ perimeter = p = 180 \\ sides = 60 \\ semi \: perimeter = s = \frac{p}{2} = \frac{180}{2} = 90\\ \\ = > \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{90(90 - 60)(90 - 60)(90 - 60)} \\ \\ = > \sqrt{90 \times 30 \times 30 \times 30 }\\ \\ = > \sqrt{30 \times 30 \times 30 \times 30 \times 3} \\ \\ = > 30 \times 30 \sqrt{3} \\ \\ = > 900 \sqrt{3}$

☆ Hope it helps ☆

Answer 2

let the side of the triangle be a. Heron's formula is as follows : Area=
$\sqrt{s(s - a)(s - b)(s - c)}$
where s= a+b+c/2 , a,b and c are respective lengths of the sides of the triangle. As in a equilateral triangle, all sides are equal => so s=a+a+a/2 =3a/2.
Area of equilateral triangle using Heron's formula=
√(1.5a)(1.5a-a)(1.5a-a)(1.5a-a) =
$\frac{ \sqrt{3 \times {a}^{2} . {a}^{2} } }{4}$
$= \frac{ \sqrt{3} {a}^{2} }{4}$
Hope it helps you...