Golden Ratio 2: A circle, a square, and Φ.

$Φ$ , or the golden ratio is $\frac{1+√5}{2}$ . Recognizing that $Φ$ is in the answer for the area, and to obtain the radius you have to take the principal root of the area, we should perform $\sqrt{Φ}$ . By simplification and radicalizing the denominator, $\sqrt{Φ} =$ $\frac{√(2+√20)}{2}$ . Since the circumference is $2rπ$ , where $r$ is the radius, we can divide the circumference length by $2π$ to obtain the radius. However, it is a good idea to keep the expression $\frac{√(2+√20)}{2}$ so we can obtain $π$ in our area. Dividing by $2π$ gives us $4$ $\frac{√(2+√20)}{2}$ , and substituting this for the formula for the area of a circle, $πr^2$ , we get the area to be 16π $\frac{1+√5}{2}$ . Since $\frac{1+√5}{2}$ $= Φ$ , our answer is in the form $aπΦ$ , where $a=16$ . The question asks for $a^2$ , so $a^2=16^2=256$ , which is our final answer.