Area of an Equilateral Triangle using Heron's Formula

To find the area of an equilateral triangle using Heron's formula, we need to know the length of one side of the triangle. Let's assume the length of each side is 'a' units.

Heron's Formula:
Heron's Formula is used to find the area of a triangle when the lengths of all three sides are given. The formula is as follows:

Area = √[s(s-a)(s-a)(s-a)]

where s is the semi-perimeter of the triangle, which is calculated as:

s = (a + b + c) / 2

In the case of an equilateral triangle, all three sides are equal, so we can simplify the formula as:

Area = √[s(s-a)(s-a)(s-a)] = √[s(s-a)^3]

Calculating the Area of the Equilateral Triangle:

In an equilateral triangle, all three sides are equal. Therefore, the semi-perimeter can be calculated as:

s = (a + a + a) / 2 = 3a / 2

Substituting this value into the formula, we get:

Area = √[(3a/2)((3a/2)-a)((3a/2)-a)((3a/2)-a)] = √[(3a/2)(a/2)(a/2)(a/2)] = √(3a^2/4) = (a√3)/2

So, the area of the equilateral triangle is (a√3)/2 square units.

Percentage Increase in Area when Side is Doubled:

Let's assume the original length of each side is 'a' units. If we double the length of each side, the new length of each side will be '2a' units.

The area of the new triangle can be calculated using the same formula as before:

New Area = (2a√3)/2 = a√3

To find the percentage increase in area, we can use the following formula:

Percentage Increase = (New Area - Original Area) / Original Area * 100

Substituting the values, we get:

Percentage Increase = (a√3 - (a√3)/2) / ((a√3)/2) * 100

Simplifying the expression, we get:

Percentage Increase = (2a√3 - a√3) / (a√3/2) * 100 = (a√3) / (a√3/2) * 100 = 2 * 100 = 200%

Therefore, the percentage increase in the area of the triangle when each side is doubled is 200%.