An equilateral triangle is inscribed in a circle of radius r. Express the circumference C of the circle as a function of the length x of a side of the triangle.
HINT GIVEN:
First show that r^2 = (x^2/3).
Hello,
I've attached a diagram of the triangle in a circle.
1. Use Pythagoran theorem:
$\displaystyle h^2+\left( \frac{x}{2}\right)^2=x^2 \Longleftrightarrow h^2=\frac{3}{4} \cdot x^2$
$\displaystyle r=\frac{2}{3} \cdot h \Longleftrightarrow h=\frac{3}{2} \cdot r $. Thus:
$\displaystyle \frac{9}{4} \cdot r^2= \frac{3}{4} \cdot x^2$. Solve for rē and you'll get:
$\displaystyle r^2= \frac{1}{3} \cdot x^2$. Therefore: $\displaystyle r=\frac{1}{3} \cdot x \cdot \sqrt{3}$
The perimeter of a circle is calculated by:
$\displaystyle c = 2 \cdot \pi \cdot r$. Plug in the values you know:
$\displaystyle c = 2 \cdot \pi \cdot \frac{1}{3} \cdot x \cdot \sqrt{3}= \frac{2}{3} \cdot \pi \cdot x \cdot \sqrt{3}$
EB