Quandaries
and Queries 

My name is Marilynn This is a high school question. I never had th opportunity to learn even simple algerbra in school. I have been trying to learn. I understand a circumfrance, diameter, and radius. I have learned what pie is and why it is called pie and that it is approximately 3.14. My son is frustrated that I can't help him with his math, The problem is this: A Circle is evenly divided into six equal triangles leaving an area between the outside of the circle and the one side of the triangle. This area is measured as 3.14. I think they used the number 3.14 to try to throw me off, knowing that pie equalls 3.14. What is the length of the radius, one line on the triangle? I wish I could draw you a picture. I hope I am clear in what I mean. It would mean so much to me if you could help me and it explaine it to me as simply as possible for my brain. Thank you so very much. Sincerely, 

Hi Marilynn, First, I will tell you my assumptions about what your picture looks like:
The unknown area is what is left around the outside of the circle, between the circumference and the outside side of each triangle. This unknown area is split into six congruent sections (equal in every way  like the triangles are "equal" in every way  including area, length of side and length of arc). It is not clear whether you mean the area of one arc is Pi (not actually split Pie, although pronouced the same), or all six sections taken together is Pi. In what follows, I will assume that one of the outside sections has an area of Pi, so that the area of all six sections taken together is 6 times Pi. We need to find the radius of the circle, call it r. I will also use P to stand for Pi (or approx. 3.14). Now, there is a lot of information we can garnish from the triangles:
From the geometry of the triangles, we need to do some algebra. Because we know an area, and we need a radius, we need a formula (or equation) for radius in terms of area (in what follows, keep the goal of a formula in mind). The area of the circle is P*r^{2} (r^{2} means r squared, and * means multiply) The area of the leftover sections outside the triangles, but inside the circle (an area we already no to be 6*P), can also be found by subtracting the area of the triangles from the area of the circle. We need to find an area of one of the equilateral triangles. Method 1 for calculating an area of a triangle Using trig in one of the right angled triangles, we have Method 2 for calculating an area of a triangle Recall that the area of the circle minus the area of all six triangles is the area of the leftover sections. From this we can now create an equation: P*r^{2}  6*(r^{2})*sqr(3)/4 = 6*P Some algebra is needed to solve for r: Factor r^{2} on the left side (r^{2})*(P6*sqr(3)/4) = 6*P Get r^{2} be itself by dividing: r^{2} = 6*P/(P6*sqr(3)/4) Take the square root of both sides: 

