tangent circles . .

$Area~between~three~touching~unit~circles~=\sqrt3 - \frac 1 2 \pi.\\ ~~~~\\ In ~the~circles~shown~there~are~9~such~areas. ~~ So~ area~of~each~circle~is~also~=\dfrac{9\sqrt3 - \frac {9\pi}2} 9 =\sqrt3 - \frac 1 2 \pi..\\ ~~~~\\ \implies~Each~of~the~circle~shown~is~an~unit~circle~circumference~2\pi.\\ \\ ~~~~\\ Middle~two~circles~of~each~ of~ the~ three~ sides~ contribute~ half~ circle~ that~ is~ \pi.\\ \\ ~~~~\\ So ~these ~six ~circles ~contribute ~6\pi.\\ \\ ~~~~\\ Three~ corner~ circles~ have ~\frac 1 3 \pi~ segment~ not~ on~ perimeter.~ So~ each~ has~ \frac 5 3 \pi~ contribution.~ Total~~ 3*\frac 5 3 \pi=5\pi.\\ \\ ~~~~\\ So~outside~ perimeter ~of ~the ~figure ~ =(6+5)\pi=\color{#D61F06}{11\pi}.$