The base of a right prism is an equilateral triangle. If the perimeter of the triangular base is 18 cm and the height of the prism is 20 cm, what is the volume of the prism in cubic centimetres?

Introduction / Context:
This problem involves the volume of a right prism whose base is an equilateral triangle. The question checks understanding of two formulas: the area of an equilateral triangle and the volume formula for a prism. Such questions are very common in aptitude tests under the topic of mensuration and solid geometry, where candidates must combine basic geometry formulas with careful substitution of given values.

Given Data / Assumptions:

• The base of the prism is an equilateral triangle ABC.
• The perimeter of the equilateral triangle is 18 cm.
• The height of the prism is 20 cm.
• The prism is right, meaning the lateral edges are perpendicular to the base.
• We are asked to find the volume of the prism in cubic centimetres.

Concept / Approach:
For a right prism, volume = area of base * height. Since the base is an equilateral triangle, if its side length is a, then perimeter = 3a. The area of an equilateral triangle of side a is given by (√3 / 4) * a^2. Once side length a is found from the perimeter, this area formula is used. Then the area is multiplied by the given height of the prism to obtain the volume. All steps require algebraic care but no advanced geometry beyond standard formulas.

Step-by-Step Solution:
Step 1: Let the side length of the equilateral triangle be a cm. Since the perimeter is 18 cm, 3a = 18, so a = 6 cm.Step 2: Compute the area of the equilateral triangle using area = (√3 / 4) * a^2.Step 3: Substitute a = 6 to get base area = (√3 / 4) * 6^2 = (√3 / 4) * 36 = 9√3 cm^2.Step 4: The height of the prism is h = 20 cm.Step 5: Volume of the prism = base area * height = 9√3 * 20 = 180√3 cm^3.

Verification / Alternative check:
As a quick check, note that the side length 6 cm matches the perimeter 18 cm exactly when multiplied by 3. The triangular base area 9√3 cm^2 is standard for an equilateral triangle with side 6 cm. Multiplying by height 20 cm clearly gives 180√3 cm^3. There is no other interpretation of the prism dimensions, so the computed volume is consistent and unique.

Why Other Options Are Wrong:

• 60√3 cm^3 assumes either the wrong area formula or the wrong height multiplication.
• 120√3 cm^3 usually comes from using an incorrect base area such as 6√3 cm^2 instead of 9√3 cm^2.
• 240√3 cm^3 would require a base area of 12√3 cm^2, which contradicts the perimeter information.
• 90√3 cm^3 corresponds to using half the correct height or some other partial calculation, so it underestimates the volume.

Common Pitfalls:

• Forgetting that perimeter 18 cm means each side is 6 cm, not 18 cm.
• Using an incorrect formula for the area of an equilateral triangle, such as 1/2 * base * height without computing the height correctly.
• Confusing the height of the prism with the side length of the base.

Final Answer:
The volume of the prism is 180√3 cm^3.