3. A right pyramid has a square base with side length $12 \mathrm{~m}$ and a height of $7 \mathrm{~m}$. Calculate the surface area of the pyramid to the nearest square metre.
Answer
Round the surface area to the nearest square meter: \(\boxed{365 \mathrm{m^2}}\)
Step 1 ：Calculate the area of the square base: \(A_{base} = 12^2 = 144 \mathrm{m^2}\)
Step 2 ：Use the Pythagorean theorem to find the slant height: \(l = \sqrt{7^2 + (\frac{12}{2})^2} \approx 9.22 \mathrm{m}\)
Step 3 ：Calculate the area of one triangular face: \(A_{triangle} = \frac{1}{2} \cdot 12 \cdot 9.22 \approx 55.32 \mathrm{m^2}\)
Step 4 ：Calculate the total surface area: \(A_{total} = A_{base} + 4A_{triangle} = 144 + 4(55.32) \approx 365.27 \mathrm{m^2}\)
Step 5 ：Round the surface area to the nearest square meter: \(\boxed{365 \mathrm{m^2}}\)
