1. Problem: Calculate the surface area of pyramids. For a square-based pyramid: - Base area = side \times side = $10 \times 10 = 100$ m$^2$ - Each triangular face area = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times 8 = 40$ m$^2$ - Total triangular faces area = $4 \times 40 = 160$ m$^2$ - Total surface area = base area + triangular faces area = $100 + 160 = 260$ m$^2$ For a triangular-based pyramid: - Each triangular face area = $\frac{1}{2} \times 20 \times 17 = 170$ cm$^2$ - Total surface area = $4 \times 170 = 680$ cm$^2$ 2. Problem: Calculate the surface area of prisms. For the rectangular prism: - Front and back area = $2 \times (4 \times 12) = 2 \times 48 = 96$ cm$^2$ - Sides area = $2 \times (4 \times 10) = 2 \times 40 = 80$ cm$^2$ - Top and base area = $2 \times (12 \times 10) = 2 \times 120 = 240$ cm$^2$ - Total surface area = $96 + 80 + 240 = 416$ cm$^2$ For the triangular prism: - Triangle ends area = $2 \times \left(\frac{1}{2} \times 24 \times 5\right) = 2 \times 60 = 120$ cm$^2$ - Sides area = $2 \times (13 \times 8) = 2 \times 104 = 208$ cm$^2$ - Base area = $24 \times 8 = 192$ cm$^2$ - Total surface area = $120 + 208 + 192 = 520$ cm$^2$ For the triangular prism (c): - Triangle ends area = $2 \times \left(\frac{1}{2} \times 12 \times 9\right) = 2 \times 54 = 108$ cm$^2$ - Sloping face area = $15 \times 12.5 = 187.5$ cm$^2$ - Back face area = $9 \times 12.5 = 112.5$ cm$^2$ - Base area = $12 \times 12.5 = 150$ cm$^2$ - Total surface area = $108 + 187.5 + 112.5 + 150 = 558$ cm$^2$ 3. Problem: Calculate the surface area of a cylinder with radius 3 cm and height 8 cm. - Area of circle = $\pi r^2 = \pi \times 3^2 = 9\pi \approx 28.27$ cm$^2$ - Circumference of circle = $\pi d = \pi \times 6 = 6\pi \approx 18.85$ cm - Area of rectangle (side) = circumference \times height = $18.85 \times 8 = 150.8$ cm$^2$ - Total surface area = $2 \times 28.27 + 150.8 = 56.54 + 150.8 = 207.34$ cm$^2$ 4. Practice: Surface area of cylinders (to 1 d.p.) a) radius 4 cm, height 15 cm - Area of circle = $\pi \times 4^2 = 16\pi \approx 50.27$ cm$^2$ - Circumference = $2\pi r = 2\pi \times 4 = 8\pi \approx 25.13$ cm - Side area = $25.13 \times 15 = 376.99$ cm$^2$ - Total surface area = $2 \times 50.27 + 376.99 = 477.53$ cm$^2$ b) radius 2.8 cm, height 20 cm - Area of circle = $\pi \times 2.8^2 = 7.84\pi \approx 24.63$ cm$^2$ - Circumference = $2\pi \times 2.8 = 5.6\pi \approx 17.59$ cm - Side area = $17.59 \times 20 = 351.84$ cm$^2$ - Total surface area = $2 \times 24.63 + 351.84 = 401.10$ cm$^2$ c) radius 30 mm (3 cm), height 15 mm (1.5 cm) - Area of circle = $\pi \times 3^2 = 9\pi \approx 28.27$ cm$^2$ - Circumference = $2\pi \times 3 = 6\pi \approx 18.85$ cm - Side area = $18.85 \times 1.5 = 28.27$ cm$^2$ - Total surface area = $2 \times 28.27 + 28.27 = 84.81$ cm$^2$ 5. Problem: Compare surface area of a tetrahedron and a cylinder. - Tetrahedron faces are all equal triangles with sides 14 cm, 13 cm, and 16 cm. - Use Heron's formula for one triangle area: - Semi-perimeter $s = \frac{14 + 13 + 16}{2} = 21.5$ cm - Area $= \sqrt{s(s-14)(s-13)(s-16)} = \sqrt{21.5 \times 7.5 \times 8.5 \times 5.5} \approx 84.68$ cm$^2$ - Total tetrahedron surface area = $4 \times 84.68 = 338.72$ cm$^2$ - Cylinder radius 4 cm, height 13 cm - Area of circle = $\pi \times 4^2 = 16\pi \approx 50.27$ cm$^2$ - Circumference = $2\pi \times 4 = 8\pi \approx 25.13$ cm - Side area = $25.13 \times 13 = 326.69$ cm$^2$ - Total cylinder surface area = $2 \times 50.27 + 326.69 = 427.23$ cm$^2$ Conclusion: The cylinder has the greater surface area.