Chapter 2.6, Problem 16AYU

16. Geometry An equilateral triangle is inscribed in a circle of radius r. See the figure. Express the circumference C of the circle as a function of the length x of a side of the triangle.

[Hint: First show that $r^2\text{ = }\frac{x^2}{3}$.]

To determine

To express the circumference $C$ of a circle as function of length $x$, where $x$ is the side of an equilateral triangle inscribed in the circle.

Answer to Problem 16AYU

Solution:

$C\left(x\right)=\frac{2\pi }{\sqrt{3}}x$

Explanation of Solution

Given:

An equilateral triangle of side ‘$x$’ is inscribed inside a circle of radius ‘$r$’ as given below.

Calculation:

To express the circumference $C$ of a circle as function of length $x$, where $x$ is the side of an equilateral triangle inscribed in the circle.

When an equilateral triangle is inscribed inside a circle, then the centre of the circle matches with the centroid of the triangle.

If $x=$ side of the triangle then, height of the equilateral triangle $=\frac{\sqrt{3}}{2}x$.

Now, centroid of a triangle divides the height in the ratio 2:1.

Therefore, the distance between centroid to any vertices of the equilateral triangle $=2/3$ of height of the equilateral triangle.

$r=\frac{2}{3}\times \left(\frac{\sqrt{3}}{2}x\right)$

$r=\frac{1}{\sqrt{3}}x$

The circumference of the circle is $C\left(x\right)=2\pi r=\frac{2\pi }{\sqrt{3}}x$.