A right prism has a regular hexagonal base with side length 6 cm. If the total surface area of the prism is 216√3 cm2, what is the height of the prism in centimetres?

Introduction / Context:
This question focuses on the surface area of a right prism with a regular hexagonal base. The base side length and the total surface area are known, and the task is to find the height of the prism. It combines area formula for a regular hexagon with the general surface area formula for a prism, which is an essential combination in solid geometry and aptitude exams.

Given Data / Assumptions:

• The base is a regular hexagon with side length a = 6 cm.
• Total surface area (TSA) of the prism = 216√3 cm2.
• The prism is right, so lateral faces are rectangles with height equal to the prism height.
• We need the height h of the prism.

Concept / Approach:
For a right prism, total surface area T is given by T = 2 * (area of base) + (perimeter of base) * height. For a regular hexagon of side a, the perimeter P = 6a, and the area A_base = (3√3 / 2) * a^2. After computing base area and perimeter using a = 6 cm, substitute into the TSA expression and solve the resulting linear equation for h. This direct substitution approach avoids unnecessary complications.

Step-by-Step Solution:
Step 1: For a regular hexagon of side a, area A_base = (3√3 / 2) * a^2 and perimeter P = 6a.Step 2: Substitute a = 6 cm. Then P = 6 * 6 = 36 cm.Step 3: Compute base area: A_base = (3√3 / 2) * 6^2 = (3√3 / 2) * 36 = 54√3 cm2.Step 4: Total surface area T = 2 * A_base + P * h = 2 * 54√3 + 36h = 108√3 + 36h.Step 5: Given T = 216√3, set 108√3 + 36h = 216√3.Step 6: Subtract 108√3 from both sides to get 36h = 108√3, so h = (108√3) / 36 = 3√3 cm.

Verification / Alternative check:
Substitute h = 3√3 back into the TSA formula. T = 108√3 + 36 * 3√3 = 108√3 + 108√3 = 216√3 cm2, which matches the given total surface area. Thus the height value is consistent and unique for the given base dimensions and total surface area.

Why Other Options Are Wrong:

• 6√3 cm and 6 cm correspond to different TSA values and do not satisfy T = 216√3 cm2 when substituted.
• 3 cm is too small; it would give much less surface area than required.
• 9 cm produces a TSA larger than 216√3 cm2, so it is not correct either.

Common Pitfalls:

• Using a wrong formula for area of a regular hexagon, such as treating it as a simple rectangle or triangle.
• Forgetting the factor of 2 multiplying the base area in the total surface area formula.
• Making algebra mistakes when solving the linear equation for h.

Final Answer:
The height of the prism is 3√3 cm.