Hello, struck!

An equilateral triangle with sides of 8cm length each,

is inscribed in a circle. Calculate the diameter of the circle. Code:

A
* * *
* /|\ *
* / | \ *
* / | \ *
/ | \ 8
* / | \ *
* / O* r \ *
* / | * \ *
/ | * \
B*- - - - + - - - -*C
* D 4 *
* *
* * *

Equilateral $\displaystyle \Delta ABC\!:\;\;AB \,=\,BC\,=\,CA\,=\,8$

It is inscribed in circle $\displaystyle O$ with radius $\displaystyle r\!:\;OC \,=\,r$

In right triangle $\displaystyle ADC\!:\;\;AD^2 + 4^2 \:=\:8^2\quad\Rightarrow\quad AD \:=\:4\sqrt{3}$

The medians, altitudes and angle bisectors intersect at $\displaystyle O$,

. . the centroid, which divides the altitude in the ratio 1:2.

Hence: .$\displaystyle OD \:=\:\frac{1}{3}\left(4\sqrt{3}\right) \:=\:\frac{4\sqrt{3}}{3}$

In right triangle $\displaystyle ODC\!:\;r^2 \:=\:CD^2 + OD^2 \;=\;4^2 + \left(\frac{4\sqrt{3}}{3}\right)^2$

[Remember: they asked for the *diameter*.]