Inscribed Circle (Inradius) – Triangle Area, Circle Area, Perimeter & Ratio
Compute inradius 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 – Inscribed circle
Inradius (from three sides): \[ r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}},\quad s=\frac{a+b+c}{2}\]
Triangle area (Heron): \[ A_t = \sqrt{s(s-a)(s-b)(s-c)} = r \cdot s \]
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) — inradius
\[ s = \frac{6+7+10}{2}=11.5 \]
\[ A_t = \sqrt{11.5(11.5-6)(11.5-7)(11.5-10)} = \sqrt{11.5 \times 5.5 \times 4.5 \times 1.5} = \sqrt{427.6875} \approx 20.680\]
\[ r = \sqrt{\frac{(11.5-6)(11.5-7)(11.5-10)}{11.5}} = \sqrt{\frac{5.5 \times 4.5 \times 1.5}{11.5}} = \sqrt{\frac{37.125}{11.5}} = \sqrt{3.22826} \approx 1.797\]
\[ A_c = \pi r^2 \approx \pi \times 3.229 \approx 10.14,\quad P_c = 2\pi r \approx 11.29\]
\[ \rho = \frac{A_c}{A_t} \approx \frac{10.14}{20.680} \approx 0.490\]
All values shown with 3‑decimal precision by default. Change decimal places above.
Incircle (Wikipedia) | Circumcircle tutorial | All geometry calculators