Question

derive the formula for area of an equilateral triangle from heron's formula​

Answer 1

Answer:

Area of equilateral triangle is √3/4

Step-by-step explanation:

By herons' formula= √(3a/2-a)(3a/2-a)(3a/2-a)

area=√3a/4a.

Answer 2

Derivation of area of an equilateral triangle through the Heron’s Formula:-

Heron’s Formula:-

$\sqrt{s(s-a)(s-b)(s-c)}$

where,

• s = Semi - Perimeter &
• a, b, c = Measurements of sides.

Or here,

Let one side of the equilateral triangle be = s

Hence,

Semi perimeter (s) = 3a/2

Change in heron’s formula in this case:-

$\sqrt{s(s-a)^3}$

= $\sqrt{s(s-a)^3}$

= $\sqrt{s{(s - a)}^{2}(s - a)}$

= $(s - a) \sqrt{s(s-a)}$

= $(s - a)\sqrt{{s}^{2} - as}$

Putting the values:-

= $(\frac{3a}{2} - a)[\sqrt{{(\frac{3a}{2})}^{2} - (a)(\frac{3a}{2})}]$

= $(\frac{a}{2})(\frac{a}{2} \sqrt{3})$

= $\frac{a^2}{4} \sqrt{3}$

For convenience,

= $\boxed{ \frac{\sqrt{3}}{4} a^2}$

Note: Some steps aren't showed. Try to add those step too, though they aren't essential.

Final Answer:-

Area of triangle:-

$\boxed{\huge{\frac{\sqrt{3}}{4} a^2}}$

where,

a is the measurement of one side of a triangle.