1. **Stating the problem:** We are given various shapes and measurements and asked to complete a table of areas and perimeters, and then find the perimeter and area of several composite shapes. 2. **Formulae and rules:** - Area of rectangle: $$\text{Area} = \text{length} \times \text{width}$$ - Perimeter of rectangle: $$\text{Perimeter} = 2(\text{length} + \text{width})$$ - For composite shapes, break them into rectangles or squares, find individual areas and perimeters, then sum appropriately. 3. **Complete the table:** Given some areas and lengths/widths, find missing values and perimeters. (a) Length unknown, Width = 11m, Area = 5m² $$\text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{5}{11} = \frac{5}{11}m$$ Perimeter: $$2\left(\frac{5}{11} + 11\right) = 2\left(\frac{5}{11} + \frac{121}{11}\right) = 2\times \frac{126}{11} = \frac{252}{11} \approx 22.91m$$ (b) Length unknown, Width = 8m, Area = 56m² $$\text{Length} = \frac{56}{8} = 7m$$ Perimeter: $$2(7 + 8) = 2 \times 15 = 30m$$ (c) Length unknown, Width = 25mm, Area = 25mm² $$\text{Length} = \frac{25}{25} = 1mm$$ Perimeter: $$2(1 + 25) = 2 \times 26 = 52mm$$ (d) Length unknown, Width = 6m, Area = 22m² $$\text{Length} = \frac{22}{6} = \frac{11}{3}m \approx 3.67m$$ Perimeter: $$2\left(\frac{11}{3} + 6\right) = 2\left(\frac{11}{3} + \frac{18}{3}\right) = 2 \times \frac{29}{3} = \frac{58}{3} \approx 19.33m$$ (e) Length unknown, Width = 8cm, Area = 12m² (assuming cm²) $$\text{Length} = \frac{12}{8} = 1.5cm$$ Perimeter: $$2(1.5 + 8) = 2 \times 9.5 = 19cm$$ (f) Length unknown, Width = 3cm, Area = 34mm (likely typo, assume 34cm²) $$\text{Length} = \frac{34}{3} \approx 11.33cm$$ Perimeter: $$2(11.33 + 3) = 2 \times 14.33 = 28.66cm$$ (g) Length unknown, Width = 8mm, Area = 21m² (likely typo, assume 21mm²) $$\text{Length} = \frac{21}{8} = 2.625mm$$ Perimeter: $$2(2.625 + 8) = 2 \times 10.625 = 21.25mm$$ (h) Length unknown, Width = 7m, Area = 50m² $$\text{Length} = \frac{50}{7} \approx 7.14m$$ Perimeter: $$2(7.14 + 7) = 2 \times 14.14 = 28.28m$$ 4. **Find perimeter and area of shapes:** (a) L-shaped figure (all lengths in cm): - Break into two rectangles: - Rectangle 1: 3cm by 7cm - Rectangle 2: 3cm by 4cm (since 7 - 3 = 4) - Area: $$3 \times 7 + 3 \times 4 = 21 + 12 = 33cm^2$$ - Perimeter: sum of outer edges: $$3 + 7 + 3 + 4 + 6 + 4 = 27cm$$ (b) U-shaped figure (all lengths in m): - Outer rectangle: height 8m, bottom width 3m + 3m = 6m - Inner rectangle cutout: 2m by 4m - Area: $$8 \times 6 - 2 \times 4 = 48 - 8 = 40m^2$$ - Perimeter: sum outer edges minus inner edges counted twice: $$8 + 3 + 3 + 8 + 3 + 3 = 28m$$ (c) Stepped shape (all lengths in m): - Break into rectangles: - Left rectangle: 5m by 2m - Bottom rectangle: 2m by 2m - Right rectangle: 7m by 6m - Area: $$5 \times 2 + 2 \times 2 + 7 \times 6 = 10 + 4 + 42 = 56m^2$$ - Perimeter: sum all outer edges: $$5 + 2 + 2 + 6 + 7 + 5 = 27m$$ (d) Stepped rectangle with top protrusion (all lengths in cm): - Total width: 18cm - Heights: 4cm, 4cm, 4cm, 15cm - Area: sum rectangles $$4 \times 18 + 4 \times 4 = 72 + 16 = 88cm^2$$ - Perimeter: sum outer edges $$4 + 18 + 15 + 18 + 4 = 59cm$$ (e) H-like shape (all lengths in cm): - Break into rectangles: - Left vertical: 6cm by 1cm - Middle vertical: 3cm by 1cm - Right vertical: 6cm by 1cm - Top horizontal: 9cm by 1cm - Area: $$6 \times 1 + 3 \times 1 + 6 \times 1 + 9 \times 1 = 6 + 3 + 6 + 9 = 24cm^2$$ - Perimeter: sum outer edges $$6 + 9 + 6 + 3 + 6 + 9 = 39cm$$ (f) T-shaped figure (all lengths in m): - Bottom horizontal: 10m - Top vertical stem: 2m wide, 6m tall - Left top horizontal: 4m - Right vertical: 3m - Area: $$10 \times 2 + 4 \times 6 = 20 + 24 = 44m^2$$ - Perimeter: sum outer edges $$10 + 6 + 2 + 4 + 3 + 6 = 31m$$