1. **Stating the problem:** We have a regular hexagon with side length $16$ and an inscribed right triangle sharing one side of the hexagon as its base. The triangle's height is $10$. We need to find: - Area of the outside (hexagon) - Area of the inside (triangle) - Area of the shaded region (difference between hexagon and triangle) 2. **Formula for the area of a regular hexagon:** The area $A$ of a regular hexagon with side length $s$ is given by: $$A = \frac{3\sqrt{3}}{2} s^2$$ 3. **Calculate the area of the hexagon:** Substitute $s=16$: $$A_{hex} = \frac{3\sqrt{3}}{2} \times 16^2 = \frac{3\sqrt{3}}{2} \times 256$$ 4. **Formula for the area of a triangle:** The area $A$ of a triangle is: $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ 5. **Calculate the area of the triangle:** Base $=16$, height $=10$: $$A_{tri} = \frac{1}{2} \times 16 \times 10 = 8 \times 10 = 80$$ 6. **Calculate the area of the shaded region:** This is the hexagon area minus the triangle area: $$A_{shaded} = A_{hex} - A_{tri} = \frac{3\sqrt{3}}{2} \times 256 - 80$$ 7. **Simplify the hexagon area:** $$\frac{3\sqrt{3}}{2} \times 256 = 3\sqrt{3} \times 128 = 384\sqrt{3}$$ 8. **Final answers:** - Area of outside (hexagon): $384\sqrt{3}$ - Area of inside (triangle): $80$ - Area of shaded region: $384\sqrt{3} - 80$