Circumcircle: Triangle Area, Circle Area, Perimeter & Ratio (purple theme)
Compute circumradius R, triangle area At, circle area Ac, circle perimeter (circumference) Pc, and the ratio Ac / At from three sides a, b, c. Decimals allowed, selectable precision.
Key formulas
Circumradius (from three sides): \[ R = \frac{a b c}{4\sqrt{s(s-a)(s-b)(s-c)}},\quad s=\frac{a+b+c}{2}\]
Triangle area (Heron): \[ A_t = \sqrt{s(s-a)(s-b)(s-c)} \]
Circle area: \[ A_c = \pi R^2 \] Circle perimeter (circumference): \[ P_c = 2\pi R \]
Ratio: \[ \rho = \frac{A_c}{A_t} \]
Enter positive real numbers, respecting triangle inequality. Choose decimal places for output.
Worked example (sides 6, 7, 10) — with LaTeX
\[ s = \frac{6+7+10}{2}=11.5 \]
\[ A_t = \sqrt{11.5(11.5-6)(11.5-7)(11.5-10)} = \sqrt{11.5 \times 5.5 \times 4.5 \times 1.5} = \sqrt{427.6875} \approx 20.680\]
\[ R = \frac{6\cdot7\cdot10}{4\,A_t} = \frac{420}{4 \times 20.680} = \frac{420}{82.72} \approx 5.077\]
\[ A_c = \pi R^2 \approx \pi \times 25.776 \approx 80.98,\quad P_c = 2\pi R \approx 31.90\]
\[ \rho = \frac{A_c}{A_t} \approx \frac{80.98}{20.680} \approx 3.916\]
All values shown with 3‑decimal precision by default. Change decimal places above.