Question

An equilateral triangle has an apothem of 5 cm. Find the perimeter of the triangle to the nearest centimeter.

Answer 1

The apothem is the distance from the center to the midpoint of one of the sides of a regular polygon.  You can make a right triangle with the apothem, the line from the midpoint to the corner and the line from the center to the corner.  An equilateral triangle has 60 degree angles (180/3).  The right triangle has half of one of those angles so 30 degrees.  Now we have a 30-60-90 triangle where the short leg is 5cm.  The long leg, which is also half of one side of the triangle is thus $5\sqrt{3}$.  A whole side of the triangle is $10\sqrt{3}$.  Multiply that by 3 to get the perimeter of $30\sqrt{3}$.