Prisms To cover it completely with coloured paper, how much paper do we need? We want the total surface area of the hexagonal prism. For that, we must add the lateral surface area and the base areas. To calculate the lateral surface area, we need the perimeter of the hexagon; and for that we have to calculate the sides of a triangle. For any prism, the lateral surface area divided by the height gives the base perimeter. So the base perimeter of the base of a triangular prism in the problem is 48 ÷ 4 = 12 centimetres. Since the base is an equilateral triangle, the base perimeter is three times the length of a side. So, the length of a side is 12 ÷ 3 = 4 centimetres. 4 cm 4 cm 4 cm 4 cm 4 cm 4 cm The perimeter of a regular hexagon of side 4 centimetres is 6 × 4 = 24 centimetres. Since the height is also 4 centimetres, the lateral surface area is 24 × 4 = 96 square centimetres. Next, we must add the areas of both bases. The area of one triangular base is 3 × 42 = 4 3 square centimetres. 4 The area of the hexagon formed by six of these is 6 × 4 3 = 24 3 square centimetres. So, the total surface area of the hexagonal prism is 96 + (2 × 24 3) = 96 + 48 3 = 48 (2 + 3) square centimetres. If we take 1.73 as an approximation for 3 , we can see this to be a little more than 179 square centimetres. Anyway, 180 square centimetres of paper would be enough. 173
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