1. **State the problem:** Calculate the surface area of a prism whose cross-section is an isosceles triangle with a rectangular hole inside and an extension on the right side. 2. **Identify given dimensions:** - Height of large triangle $h = 25$ cm - Base of large triangle $b = 52$ cm - Rectangular hole dimensions: height $h_r = 6$ cm, width $w_r = 8$ cm - Extension on right side: lengths $36$ cm and $12$ cm 3. **Calculate area of the large isosceles triangle:** $$\text{Area}_{\triangle} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 52 \times 25 = 650 \text{ cm}^2$$ 4. **Calculate area of the rectangular hole:** $$\text{Area}_{\text{hole}} = h_r \times w_r = 6 \times 8 = 48 \text{ cm}^2$$ 5. **Calculate area of the trapezoidal extension:** The trapezoid has parallel sides $12$ cm and $52$ cm (base of triangle plus extension), and height $36$ cm. $$\text{Area}_{\text{trap}} = \frac{1}{2} \times (b + 12) \times 36 = \frac{1}{2} \times (52 + 12) \times 36 = \frac{1}{2} \times 64 \times 36 = 1152 \text{ cm}^2$$ 6. **Calculate net cross-sectional area:** $$\text{Area}_{\text{net}} = \text{Area}_{\triangle} + \text{Area}_{\text{trap}} - \text{Area}_{\text{hole}} = 650 + 1152 - 48 = 1754 \text{ cm}^2$$ 7. **Calculate surface area of the prism:** Assuming the prism length is the extension length $36$ cm (or if length is given, use that), the surface area includes: - Two cross-sectional areas (front and back): $2 \times 1754 = 3508$ cm$^2$ - Lateral surface area: perimeter of cross-section $\times$ length 8. **Calculate perimeter of cross-section:** - Triangle sides: base $52$ cm, two equal sides $s$ where $$s = \sqrt{\left(\frac{b}{2}\right)^2 + h^2} = \sqrt{26^2 + 25^2} = \sqrt{676 + 625} = \sqrt{1301} \approx 36.06 \text{ cm}$$ - Triangle perimeter: $52 + 2 \times 36.06 = 124.12$ cm - Subtract hole perimeter (not part of outer perimeter) - Add extension sides: $36 + 12 = 48$ cm - Total perimeter $= 124.12 + 48 = 172.12$ cm 9. **Calculate lateral surface area:** $$\text{Lateral area} = \text{perimeter} \times \text{length} = 172.12 \times 36 = 6196.32 \text{ cm}^2$$ 10. **Calculate total surface area:** $$\text{Surface area} = 2 \times \text{cross-sectional area} + \text{lateral area} = 3508 + 6196.32 = 9704.32 \text{ cm}^2$$ **Final answer:** The surface area of the prism is approximately **9704.32 cm²**.