Diameter=2xradius!!!!!

According to Sine rule, in $\triangle ABC, \; \left( R = \textbf{Circumradius} \right)$ $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ So the diameter is, $2R = \frac{AB}{\sin \angle C} = \frac{30}{\sin 45^{o}}$ $\Rightarrow 2R = 30\sqrt{2} \approx \boxed{42.426}$

The angle at the center is 2 * 45=90.
Isosceles triangle with leg=30 give base=42.4264 also =diameter.

quite easily done.

CAUTION: This is the first time I am posting a solution.....and I am unable to "comprehend" the "Formatting guide"......so please bare with me... I may not be able to present it in a nice manner....SORRY.!

In ABC, we know that AB=30cm , and C=45 .

In a circle, we know that the * angle subtended by a chord on the circle *, on any point on the circle is equal (unless the point is on the same segment...)

With the given data about the triangle and the above given theorem, let us assume that AB as the chord, and its circumcircle as the circle.

If the chord (AB) subtends 45 on the circle, then it must subtend 90 at the centre of the circle.We also know that tahe angle subtended by a chord at the centre is twice the angle subtended by it on the circle.

Then, assuming the centre to be O,

                                         angle(AOB) = 2 [angle(ACB)]= 90

Which implies, in AOB,

                                   angle (AOB)=90 ; ab=30cm ; AO=OB { radius }

With this we come to know that , AO=OB=15{2}

Therefore, diameter = AO + OB = 30{2}