A right pyramid is on a regular hexagonal base. Each side of the base is $10\text{ m}$ and the height is $60\text{ m}$. The volume of the pyramid is

### Concept & Formula The area of a regular hexagon can be found by treating it as six equilateral triangles. $$ \text{Base Area} = 6 \times \frac{\sqrt{3}}{4} \times a^2 $$ $$ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times h $$ ### Step-by-Step Solution 1. **Given:** - Side of hexagonal base, $a = 10\text{ m}$ - Height, $h = 60\text{ m}$ 2. **Calculate Base Area:** - $\text{Base Area} = 6 \times \frac{\sqrt{3}}{4} \times (10)^2$ - $\text{Base Area} = 6 \times \frac{\sqrt{3}}{4} \times 100 = 6 \times 25\sqrt{3} = 150\sqrt{3}\text{ m}^2$ 3. **Calculate Volume:** - $\text{Volume} = \frac{1}{3} \times 150\sqrt{3} \times 60$ - $\text{Volume} = 50\sqrt{3} \times 60 = 3000\sqrt{3}$ 4. **Convert to Decimal:** - Since $\sqrt{3} \approx 1.732$ - $\text{Volume} = 3000 \times 1.732 = 5196\text{ m}^3$ ### Exam Strategy & Shortcut Instead of multiplying large numbers, group them logically: $\frac{1}{3} \times 60 = 20$. Then $20 \times 150\sqrt{3} = 3000\sqrt{3}$. You can mentally shift the decimal place of $1.732$ three places to the right to get $1732$, and then multiply by $3$ to get $5196$. ### Common Pitfall A common mistake is calculating the perimeter instead of the area of the hexagon, or forgetting the $\frac{1}{3}$ in the pyramid volume formula. ### Final Answer Therefore, the correct answer is **$5196\text{ m}^3$**.