O Cambridge University Press \& Assessment 2022 [Tur \( \mathbf{1 9} \) The cross-section of a prism is an equilateral triangle of side 6 cm . The length of the prism is 20 cm . Calculate the total surface area of the prism.

To find the total surface area of the prism, we first calculate the area of the triangular base. The area \( A \) of an equilateral triangle can be found using the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \] where \( s \) is the side length. Plugging in \( s = 6 \) cm: \[ A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \, \text{cm}^2 \] Since the prism has two triangular bases, the total area of the bases is: \[ 2A = 2 \times 9\sqrt{3} = 18\sqrt{3} \, \text{cm}^2 \] Next, we find the area of the three rectangular sides. The perimeter \( P \) of the triangular base is: \[ P = 3 \times 6 = 18 \, \text{cm} \] The area of the three rectangular sides is given by: \[ \text{Area of rectangles} = \text{Perimeter} \times \text{Length} = 18 \times 20 = 360 \, \text{cm}^2 \] Finally, we sum the areas of the bases and the rectangular sides to find the total surface area \( SA \): \[ SA = \text{Area of bases} + \text{Area of rectangles} = 18\sqrt{3} + 360 \, \text{cm}^2 \] Thus, the total surface area of the prism is: \[ SA \approx 18 \times 1.732 + 360 \approx 31.176 + 360 \approx 391.176 \, \text{cm}^2 \] So the total surface area is approximately \( 391.18 \, \text{cm}^2 \).