Points to remember $\Rightarrow$

• Regular hexagon with perimeter of 21, has side length of 3.5, because 21/6 = 3.5

• Radius of the circle is 3.5, as it is equal to the side length of the hexagon.

• In radians, arc length of a circle = radius times the angle, written as $r \theta$ .

• All angles of a regular hexagon are equal to 120 degrees or 2.0944 radians

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$Perimeter_{cut \ pizza} \ = \ 2r \ + \ Perimeter_{full \ pizza} \ - \ Cut \ Arc \ Length$

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$\begin{aligned} Perimeter_{full \ pizza} \ &= \ 2 \pi r \\ &= \ 2 \times \pi \times \ 3.5 \\ &= 21.9911485751 \ldots \end{aligned}$

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$\begin{aligned} Cut \ Arc \ Length \ &= r \theta \\ &= radius \times angle \ subtending \ arc \\ &= 3.5 \times 2.0944 \ Radians \\ &= 7.3304\end{aligned}$

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$\begin{aligned} 2r \ + \ Perimeter_{full \ pizza} \ &- \ Cut \ Arc \ Length \\ &= 7 + 21.9911485751 - 7.3304 \\ &= 21.66074 \ldots \\ &\approx \boxed{\underline{21.66}} \end{aligned}$

Given that the regular hexagon has a perimeter of 21, we find that the length of one of the sides is $\frac{21}{6} = \frac{7}{2}$ . This is useful because this side length is also the radius of the circle.

Now, we can use this radius to work out the pizza's circular perimeter. Since there is only $\frac{2}{3}$ of the pizza left, we need to find $\frac{2}{3}$ of the total circular perimeter: $2\left(\frac{7}{2}\right)\pi \times \frac{2}{3} = \frac{14}{3}\pi$

Finally, to get the full perimeter of the sliced pizza, we have to add 2 radii:

$\frac{14}{3}\pi + 2\left(\frac{7}{2}\right) = \boxed{21.66}$