How to find the remaining area of equilateral triangle if 3 circle sectors of radius R1, R2, R3 are given.

Question

An equilateral triangle of side length $S$ is given. and from the three corners of the triangle, three circle sector is being drawn which has a radius of $R_1$, $R_2$, $R_3$. Find the area of the triangle which is not covered by any of these circle sectors.

So if

$R_1+R_2 < S$ and $R_2+R_3 < S$ and $R_3+R_1 < S$

then we can find the circle sector area using formula and then we can subtract it from triangle area to get the answer. but I'm stuck at the point where these sectors intersect/overlap! how to get the answer if the sector intersects/overlaps like the above figure?

Answer 1

Let $V_1,V_2,V_3$ be the vertices of the triangle at which the sectors of radii $R_1,R_2,R_3$, respectively, are centered. Let $P$ be the point of intersection, inside the triangle, of the sectors with radii $R_2$ and $R_3$. Let $T$ be the point on the side $V_2V_3$ such that $PT$ is perpendicular to $V_2V_3$. Then you can decompose the overlapping area of those two sectors as:

• a sector of radius $R_2$ centered at $V_2$, but extending from the side $V_1V_2$ only as far as $P$;
• the right triangle $V_2PT$;
• a sector of radius $R_3$ centered at $V_3$, but extending from the side $V_1V_3$ only as far as $P$;
• the right triangle $V_3PT$.

Once you know the length of $PT$ (which should be possible given $R_2$, $R_3$, and $S$), you will be able to calculate all of these areas.