1. **State the problem:** We have a regular pentagon with an inscribed circle (incircle) of radius $r=5$ cm. We need to find the area of the pentagon outside the circle, i.e., $\text{Area}_{\text{pentagon}} - \text{Area}_{\text{circle}}$. 2. **Recall formulas:** - Area of circle: $\pi r^2$ - For a regular pentagon with side length $a$ and apothem $r$, area is $\frac{1}{2} \times \text{perimeter} \times r = \frac{1}{2} \times 5a \times r$ 3. **Find side length $a$ of the pentagon:** The apothem $r$ relates to side length $a$ by $$r = \frac{a}{2 \tan(\pi/5)}$$ Rearranged: $$a = 2r \tan(\pi/5)$$ Calculate $\tan(\pi/5)$: $$\tan(36^\circ) \approx 0.7265425$$ So, $$a = 2 \times 5 \times 0.7265425 = 7.265425$$ cm 4. **Calculate area of pentagon:** $$\text{Area}_{\text{pentagon}} = \frac{1}{2} \times 5a \times r = \frac{1}{2} \times 5 \times 7.265425 \times 5 = \frac{1}{2} \times 5 \times 7.265425 \times 5 = 90.8178$$ cm$^2$ 5. **Calculate area of circle:** $$\text{Area}_{\text{circle}} = \pi r^2 = \pi \times 5^2 = 25\pi \approx 78.5398$$ cm$^2$ 6. **Calculate area outside the circle:** $$\text{Area}_{\text{outside}} = 90.8178 - 78.5398 = 12.278$$ cm$^2$ 7. **Round to 2 decimal places:** $$12.28$$ cm$^2$ **Final answer:** The area of the pentagon outside the circle is approximately $12.28$ cm$^2$.