The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? Express your answer in terms of pi and simplest radical form.

anomy
Aug 13, 2018

#1**+3 **

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?

**perimeter \(\triangle\) = 3a**

**radius **O** = r**

\(cos(30°)=\frac{a}{2r}\\ r=\frac{a}{2\cdot cos(30°)}\\ a=2r\cdot cos(30°)\)

\(\large\color{blue}3a=r^2\pi\)

\(3a=(\frac{a}{2\cdot cos(30°)})^2\cdot \pi\\ 3a=\frac{\pi a^2}{4\cdot cos^2(30°)}\\ \color{blue}a=\frac{3\cdot 4\cdot cos^2(30°) }{\pi}=\frac{3\cdot 4\cdot (\frac{1}{2}\sqrt{3})^2 }{\pi}=\frac{3^2}{\pi}\\ \color{blue}a=2.86479..\)

\(r=\frac{3\cdot 4\cdot cos^2(30°) }{2\pi\cdot cos(30°)}\\ \color{blue}r=\frac{6\cdot cos(30°)}{\pi}=\frac{6\cdot (\frac{1}{2}\sqrt{3})}{\pi}=\frac{3\cdot \sqrt{3}}{\pi}\\ \color{blue}r=1.65399\)

proof:

\(perimeter \triangle=area\ O\\ 3a=r^2\pi\\ 3\cdot \frac{3\cdot 4\cdot cos^2(30°) }{\pi}=\pi\cdot (\frac{6\cdot cos(30°)}{\pi})^2\\ 8.59437=8.59437\)

!

asinus
Aug 13, 2018