1. **Problem statement:** Find the total surface area of a building with a floor plan shaped like an irregular polygon (a "U" shape) and a flat roof. The building height is 8 m. 2. **Given dimensions:** - Bottom side length = 27 m - Left vertical side height = 18 m - Right vertical side height = 22 m - Top horizontal segment = 10 m - Building height = 8 m 3. **Step 1: Calculate the area of the floor plan (base area).** The floor plan is an irregular polygon shaped like a "U". We can divide it into rectangles and subtract the notch area. - Total rectangle area (outer rectangle): width = 27 m, height = 22 m (right side) $$\text{Outer area} = 27 \times 22 = 594 \text{ m}^2$$ - Notch area (the cut-out part): width = 27 - 10 = 17 m (since top segment is 10 m, the notch width is 27 - 10 = 17 m), height = 22 - 18 = 4 m (difference between right and left vertical sides) $$\text{Notch area} = 17 \times 4 = 68 \text{ m}^2$$ - Floor plan area = Outer area - Notch area $$594 - 68 = 526 \text{ m}^2$$ 4. **Step 2: Calculate the perimeter of the floor plan.** The perimeter is the sum of all outer edges: - Bottom side = 27 m - Left vertical side = 18 m - Right vertical side = 22 m - Top horizontal segments: 10 m + two vertical segments (not given explicitly, but from the notch height difference 4 m each side) + the remaining horizontal segments Since the notch is a "U" shape, the perimeter includes the two vertical notch sides (4 m each) and the horizontal segment at the bottom of the notch (27 - 10 = 17 m). So perimeter = 27 (bottom) + 18 (left) + 22 (right) + 10 (top left) + 4 (notch left vertical) + 17 (notch bottom) + 4 (notch right vertical) Calculate: $$27 + 18 + 22 + 10 + 4 + 17 + 4 = 102 \text{ m}$$ 5. **Step 3: Calculate the lateral surface area (walls).** Lateral surface area = perimeter × height $$102 \times 8 = 816 \text{ m}^2$$ 6. **Step 4: Calculate total surface area (walls + roof).** Total surface area = lateral surface area + floor plan area (roof area) $$816 + 526 = 1342 \text{ m}^2$$ **Final answer:** The surface area of the building, including its roof, is **1342 m²**.