Curved pentagram

Let $P$ be the center of the central pentagonal shape, $S$ be one of the vertices of the central pentagonal shape, $R$ and $Q$ be the center of the curved sides of the pentagonal shape adjacent to $S$ , $T$ be the tangent point between two of the unit circles (and also the intersection of $PR$ and arc $SQ$ ), and $O$ be the center of the unit circle that contains arc $SQ$ , as shown below.

Also, let $PQS$ be the area of the region bounded by segments $PS$ and $PQ$ and arc $SQ$ , and $PRS$ be the area of the region bounded by segments $PS$ and $PR$ and arc $RS$ , and $RST$ be the area of the region bounded by the segment $RT$ and arcs $RS$ and $ST$ .

Since $OT$ is the radius of a unit circle, $OT = 1$ , since $\angle OPT$ is one fifth of the central pentagonal shape, $\angle OPT = \frac{1}{5}360° = 72°$ , and since $\angle PTO$ is the angle between a tangent line and a radius of a circle, $\angle PTO = 90°$ . Solving $\triangle OPT$ and using the Pythagorean trigonometric identity gives us $\angle POT = 18°$ , $OP = \csc 72°$ and $OT = \cot 72°$ .

Since $OS$ is the radius of a unit circle, $OS = 1$ , and since $\angle QPS$ is one tenth of the central pentagonal shape, $\angle QPS = \frac{1}{10}360° = 36°$ . Then using $OP = \csc 72°$ and the law of sines on $\triangle OPS$ (and knowing that $\angle PSO$ is an obtuse angle), we get $\angle POS = \sin^{-1}(\sin 36° \csc 72°) - 36°$ .

The area $PQS$ would be the area of $\triangle OPS$ minus the area of the sector $OQS$ , so area $PQS = \frac{1}{2} \cdot 1 \cdot \csc 72° \sin \angle POS - \frac{\angle POS}{360°} \pi \cdot 1^2$ or $PQS = \frac{1}{2} \csc 72° \sin \angle POS - \frac{\angle POS}{360°} \pi$ . By symmetry, $PRS = PQS$ .

The area $RST$ would be the area of $\triangle OPT$ minus the areas of the sector $OQT$ and the areas of $PQS$ and $PRS$ , so area $RST = \frac{1}{2} \cdot 1 \cdot \cot 72° - \frac{18°}{360°} \pi - 2 \cdot PQS$ or $RST = \frac{1}{2} \cot 72° - \frac{1}{20} \pi - 2 \cdot PQS$ .

The area of the star-shaped region is then equal to $A = 10(PQS + RST) \approx \boxed{0.044}$ .