Question

Please help me find the area of the polygon

Answer 1

The area of a regular polygon is A = 1/2 ap, where a is the apothem and p is the perimeter.  You have the radius.  It's not much, but it's enough.  That given radius there serves as the hypotenuse of a right triangle.  Dropping the altitude from the center straight down is our apothem.  That's like a leg of that right triangle.  This is a regular hexagon.  That means that each of the 6 central angles measures 60 degrees (360/6=60).  With each radii measuring the same length, then each base angle measures the same as well by the isosceles triangle theorem.  Those measure 180-60=120 together, so each one measures 60.  Also, since this is an equiangular triangle, all the sides measure the same length as well.  That means that one side of this polygon measures $10 \sqrt{3}$.  We'll use that for perimeter.  Now we need the apothem still which serves as the height of the right triangle with a hypotenuse of 10 sqrt 3, side of 5 sqrt 3.  We need the other leg which we can find with Pythagorean's Theorem.  $(10 \sqrt{3}) ^{2} -(5 \sqrt{3}) ^{2}= a^{2}$ giving us that a = 15.  Now we have to find perimeter and we're good to fill in the formula.  Perimeter is 6(10 sqrt 3) which is 60 sqrt 3.  Let's fill in the formula now: $A= \frac{1}{2} (15)(60 \sqrt{3})$ which gives us an area of $450 \sqrt{3}$ or 779.42