Radius and angles from three arcs

Text

Three points divide a circle into three arcs of length $25.13$ cm, $37.70$ cm and $62.83$ cm respectively. Find the radius of the circle and the three corresponding central angles.

Solution

The sum of the arcs equals the circumference: $25.13+37.70+62.83=125.66=2\pi r\ \Longrightarrow\ r=\frac{125.66}{2\pi}\approx20\ \text{cm}.$ The central angles are proportional to the arcs ($2:3:5$, total $10$): $\alpha_1=\frac{2}{10}\cdot360^\circ=72^\circ,\quad \alpha_2=108^\circ,\quad \alpha_3=180^\circ.$ $\boxed{r=20\ \text{cm};\ \ 72^\circ,\ 108^\circ,\ 180^\circ}$