NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Volume of prisms

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Required skills (click to reveal)

A triangular prism has a triangular base with a base length of 13/5 m and a perpendicular height of 9/4 m. The prism extends 7/3 m in length between the two triangular bases.

The volume of a prism is calculated using the formula V = (area of base) × (length).

Calculate the exact volume of the prism in cubic metres, expressing your answer as a simplified fraction.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

A rectangular prism. A prism of length 4 cm, width 3 cm and height 2 cm has volume 24 cm³ = 24 mL. The perspective drawing is schematic.4 cm3 cm2 cm
A prism of length 4 cm, width 3 cm and height 2 cm has volume 24 cm³ = 24 mL. The perspective drawing is schematic.
Square the length conversion factor. A 1 m by 1 m square is 100 cm by 100 cm. Its area is 10,000 cm². The 10-by-10 grid shows 100 squares, each 10 cm by 10 cm.1 m = 100 cm1 m100 cm
A 1 m by 1 m square is 100 cm by 100 cm. Its area is 10,000 cm². The 10-by-10 grid shows 100 squares, each 10 cm by 10 cm.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The prism is a triangular prism with a triangular base. The base has a base length of 13/5 m and a perpendicular height of 9/4 m. The length of the prism (distance between the two triangular bases) is 7/3 m.
  2. The volume of a prism is given by V = (area of base) × (length). The base is a triangle, so its area is (1/2) × base × height.
  3. Substitute the base and height of the triangle into the area formula: area = (1/2) × (13/5) × (9/4).
  4. Multiply the numerators: 1 × 13 × 9 = 117. Multiply the denominators: 2 × 5 × 4 = 40. So, area = 117/40 m².
  5. Now use the volume formula: V = (area of base) × (length) = (117/40) × (7/3).
  6. Multiply the numerators: 117 × 7 = 819. Multiply the denominators: 40 × 3 = 120. So, V = 819/120 m³.
  7. Simplify the fraction 819/120. First, find the greatest common divisor (GCD) of 819 and 120.
  8. Factor 819: 819 ÷ 3 = 273, 273 ÷ 3 = 91, 91 = 7 × 13. So 819 = 3² × 7 × 13. Factor 120: 120 = 2³ × 3 × 5. The common factor is 3.
  9. Divide numerator and denominator by 3: 819 ÷ 3 = 273, 120 ÷ 3 = 40. So, V = 273/40 m³.
  10. Check if 273 and 40 have any common factors. 273 = 3 × 7 × 13, 40 = 2³ × 5. No common factors, so 273/40 is in simplest form.
  11. The volume is 273/40 cubic metres. This is an improper fraction, but it is already simplified.
  12. The exact volume of the prism is 273/40 m³. This matches the required format: a simplified fraction in cubic metres.

Reference answer: 273/40 m³

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