The equilateral triangle PQR has sides of length 8cm. A circle, centre O, passes through each of the vertices of the triangle. Find an expression for the circumference of circle.
I thought about breaking it up into sectors but I don’t know the radius.
I thought about perpendicular bisectors meeting in centre but that wouldn’t help me know the radius.
Grateful for help!
On
A circle that passes through the triangle's vertices is known as its circumcircle, and its radius is known as its circumradius $r_c$. The circumference is obviously $2\pi r_c$. The formula for the circumradius of a triangle with side lengths $a$, $b$, and $c$ is shown below:
Since it's an equilateral triangle, $a = b = c$.
First, draw $\triangle\mathit{PQR}$. The angles between each side are all $60$ degrees. The center of the circle, $O$, is exactly at the center of the triangle. Then, connect each vertex of the triangle to $O$. These segments are all $r$, the radius of the circle. Next, choose a side and draw the perpendicular bisector of the side between said side and $O$.
To make that easier to understand, I'll provide an example. Let's use segment $\mathit{PQ}$. Define the midpoint of $\mathit{PQ}$ as $M$. Segment $\mathit{MO}$ is the perpendicular bisector between $\mathit{PQ}$ and $O$.
Segments $\mathit{PM}$, $\mathit{MO}$, and $\mathit{MP}$ form a 30/60/90 triangle which relates the side length of the triangle to the radius of the circle.
Let's call the side length of the circle $s$. The length of segment $\mathit{PM}$ equals $\frac{s}{2}$ and $\angle \mathit{OPM}$ is $30$ degrees. Therefore, $\cos \angle \mathit{OPM} = \cos \frac{\pi}{6} = \frac{\mathit{PM}}{\mathit{OP}} = \frac{s}{2r}$. Rewriting, we get that: $$ r = \frac{1}{2} s \sec \frac{\pi}{6} $$
The circumference of a circle is:
$$ C = 2\pi r $$
Substituting $r$, we get:
$$ \boxed{C = \pi s \sec \frac{\pi}{6}} $$
This equation directly relates the side length of the triangle to the circumference of the circle.
In order to better understand the derivation, I drew this diagram.
In the context of your problem, you were meant to find an expression for the circumference of the circle with the side length of the triangle being $8\text{cm}$. That expression would be:
$$ \boxed{8\text{cm} \cdot \pi \sec \frac{\pi}{6}} $$