The base of a right prism is an isosceles trapezium with parallel sides measuring 10 cm and 20 cm, and non-parallel sides measuri…

Question

The base of a right prism is an isosceles trapezium with parallel sides measuring 10 cm and 20 cm, and non-parallel sides measuring 13 cm each. The volume of the prism is 3240 cm³. What is the total surface area of the prism?

1. A. 2120 cm²
2. B. 1620 cm²
3. C. 1940 cm²
4. D. 1368 cm² (Correct answer)

Correct Answer

Option D — 1368 cm²

Detailed Solution & Explanation

The correct answer is 1368 cm².

Key Points

• Find the trapezium's height first. The parallel sides differ by 20 − 10 = 10 cm, split 5 cm either side, so with a slant side of 13:
  • h = √(13² − 5²) = √(169 − 25) = √144 = 12 cm
• Base area = ½ × (10 + 20) × 12 = 180 cm².
• The prism's length follows from its volume: 3240 / 180 = 18 cm.
• Total surface area = 2 × base + perimeter × length:
  • perimeter = 10 + 20 + 13 + 13 = 56 cm
  • TSA = (2 × 180) + (56 × 18) = 360 + 1008 = 1368 cm²

Additional Information

• For any prism, volume = base area × length and lateral surface = perimeter of base × length; total surface adds the two bases.
• Dropping perpendiculars from the ends of the shorter parallel side is the standard construction for an isosceles trapezium — it produces two congruent right triangles and a rectangle.
• The 5-12-13 triple appears here; recognising it removes the square-root work entirely.
• Trapezium area is ½ × (sum of parallel sides) × height, sometimes written as the mean of the parallel sides times the height.