The equilateral triangle is not so common triangle such as isosceles or right triangles. If you get a question in test involving an equilateral triangle (all sides are equal), you might end up spending more time on it if you are not familiar with the formula.
Coming towards the formula, we will first see how the equilateral triangle is derived. You might be acquainted with 30:60:90. An equilateral triangle is a triangle in which all the three sides are equal. If all the sides are equal, then the angles opposite to the equal sides are also equal. Similarly, in an equilateral triangle when all the three sides are equal than all the three angles must be equal.
As triangle is made up of 180 degrees, therefore each angle must be equal to 180/3=60 degrees. If we cut an equilateral triangle in half, we will likely to get two 30:60:90. As you know the dimensions of a triangle i.e.
x: x √3: 2x
In this way, you can figure out the area of a triangle. This is one of the important formulae that you will be dealing with in your SAT exam.
Let’s change the situation here. Suppose the length of one side of the equilateral triangle is s. If we split the equilateral triangle in half, draw a line from top to bottom to form a right angle, we will get two equal 30:60:90 triangles. The shortest side will be equal to s/2. The height corresponding to the middle length of the 30:60:90 triangle will be equal to s√3/2.
As we are looking for the area of the equilateral triangle, so height is same and the base is 2 times s/2. Since the base of the equilateral triangle is two of the small side (keep in mind we split the equilateral triangle into two). In order to find the area of equilateral, we multiply the following
s x s √3/ 2= s² √3/ 2 divided by 2
Area of equilateral: s² √3/ 4
We have divided by 2 as the formula for the area of triangle is (bxh)/2
s x s √3
Why do we divide by 2? Remember that the formula for the area of any triangle is (b x h)/2