Let a be the apothem and P the perimeter of a regular polygon. The area of the regular polygon can be calculated by using the following formula.
A = 1/2aP
Next, let's consider an equilateral triangle with side length s.
Since △ ABC is a regular polygon, we can find its area by using the formula written before. Note that the perimeter of △ ABC is 3s.
A_(△ ABC) = 1/2a * 3s
Next, let's find the center of △ ABC by drawing two altitudes. The intersection point will be the center of △ ABC and the distance from it to any side is the apothem of the triangle.
Remember that in an equilateral triangle, each altitude coincides with the angle bisector and with the median. Because m∠ CAB = 60^(∘), we have that m∠ XAB = 30^(∘) and also AY = s2.
Notice that △ AXY is a 30^(∘)-60^(∘)-90^(∘) Triangle, and by the 30^(∘)-60^(∘)-90^(∘) Triangle Theorem, the length of the longer leg (AY) is sqrt(3) time the length of the shorter leg (XY).
