1. **Problem Statement:** We have a triangle with side lengths 3 cm, 4 cm, and 4.5 cm. We need to find the radius of the circumcircle (উল্লাপকের বৃত্তের বাহর) and draw the circumcircle. 2. **Formula and Important Rules:** The radius $R$ of the circumcircle of a triangle with sides $a$, $b$, and $c$ and area $A$ is given by: $$ R = \frac{abc}{4A} $$ The area $A$ can be found using Heron's formula: $$ A = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{where} \quad s = \frac{a+b+c}{2} $$ 3. **Calculate the semi-perimeter $s$:** $$ s = \frac{3 + 4 + 4.5}{2} = \frac{11.5}{2} = 5.75 $$ 4. **Calculate the area $A$ using Heron's formula:** $$ A = \sqrt{5.75(5.75-3)(5.75-4)(5.75-4.5)} = \sqrt{5.75 \times 2.75 \times 1.75 \times 1.25} $$ Calculate inside the root: $$ 5.75 \times 2.75 = 15.8125, \quad 1.75 \times 1.25 = 2.1875 $$ $$ A = \sqrt{15.8125 \times 2.1875} = \sqrt{34.578125} \approx 5.88 $$ 5. **Calculate the circumradius $R$:** $$ R = \frac{3 \times 4 \times 4.5}{4 \times 5.88} = \frac{54}{23.52} \approx 2.297 $$ 6. **Interpretation:** The circumradius of the triangle is approximately 2.297 cm. This is the radius of the circle passing through all three vertices of the triangle. **Final answer:** $$ \boxed{R \approx 2.3 \text{ cm}} $$