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).

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- Jan 22nd 2007, 05:34 PMsymmetryEquilateral Triangle Inscribed in Circle
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). - Jan 22nd 2007, 10:33 PMearboth
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 - Jan 23rd 2007, 02:35 AMsymmetryok
I totally get it.

Thanks for the picture.

Can you attach diagrams from now on with your reply? It is a lot easier for me to see what is happening that way.