Can the area of an equilateral triangle be determined without using Heron's Formula?
All the sides of an equilateral triangle have the same length. The area can be found without using Heron's Formula and instead by using the basic formula for the area of any triangle A = (1/2)*b*h where b is the base of the triangle and h is the height.
In an equilateral triangle with sides equal to L, the perpendicular dropped from any vertex to the opposite side, meets it at half the distance. This gives an right with one side equal to L/2 and the hypotenuse equal to L.
The height is `H^2 = L^2 - L^2/4`
=> `H = sqrt3*L/2`
The area of the triangle is `(1/2)*b*h = (1/2)*L*(sqrt 3)*L/2`
=> `sqrt 3*L^2/4`
The area of an equilateral triangle with sides L is `(sqrt 3/4)*L^2`
The area of any triangle can be determined using the formula:
`A = (side*side*sin(alpha))/2`
`alpha` is the included angle
The equilateral triangle has the next properties:
- the measure of all its interior angles is of 60 degrees;
- the lengths of all its sides are equal.
Use the next notation of the length of the side: side = a.
`A = (a*a*sin(60))/2`
`` `A = a^2*(sqrt3/2)/2` => `A = (a^2*sqrt3)/4`
The area of the equilateral triangle is `A = (a^2*sqrt3)/4.`