The equilateral triangle is not as common as isosceles right triangle (45:45:90 triangles) or right triangles in general. But when you see a problem involving the area of an equilateral triangle—one in which all sides are equal—you’ll likely end up spending more time on it than you would had you known the formula for the area of the equilateral triangle.
Before I just tell you the formula, let’s see how it is derived based on a triangle that might be more familiar to you: the 30:60:90 triangle. Why this triangle? Well, an equilateral triangle is a triangle in which all sides are equal. When we have equal sides the angles opposite the equal sides are always the same. When all three sides are equal, then all three angles of the triangle must be equal. A triangle is made up of 180 total degrees so each angle measure must be 180/3 = 60 degrees.
If we cut an equilateral triangle in half, we end up with two 30:60:90 triangles. Since you know the dimensions of a 30:60:90 triangle—x: x√3: 2x—you can figure out the area of the triangle this way. What? I know such thing! Actually, you do. At the front of each SAT math section, there is a helpful list of geometry formulas, including the one for a 30:60:90 triangle.
But I’m going to throw a little curveball at you. Let’s call the length of one side of the equilateral triangle s. That means that if split this equilateral triangle in half, drawing a line down from the top of the height to form a right angle with the base, we get two equal 30:60:90 triangles in which the shortest side is equal to s/2. The height, which corresponds to the middle leg of the 30:60:90 triangle, is equal to (s√3)/2.
Remember, though, we are looking for the area of the equilateral, not the 30:60:90 triangle. So while the height is the same, the base is 2 times s/2 , since the base of the equilateral is two of the smallest side of the 30:60:90 triangles (remember, we split the equilateral in two equal halves). So to find the area of the equilateral, we multiply the following:
Why do we divide by 2? Remember that the formula for the area of any triangle is (b x h)/2