Find the total surface area, in square centimetres, of a solid hemisphere with diameter 42 centimetres.

Introduction / Context:
This question focuses on the total surface area of a solid hemisphere, which includes both the curved surface area and the area of the circular base. Such calculations are frequently used in mensuration problems about bowls, domes, and other half spherical objects.

Given Data / Assumptions:

• The solid is a hemisphere.
• Diameter = 42 cm, so radius r = 21 cm.
• We must find the total surface area in square centimetres.
• Use pi = 22 / 7.

Concept / Approach:
For a hemisphere:
Curved surface area (CSA) = 2 * pi * r^2.
Area of the circular base = pi * r^2.
Total surface area (TSA) = CSA + base area = 3 * pi * r^2.
We simply substitute r = 21 cm and pi = 22 / 7 into TSA = 3 * pi * r^2.

Step-by-Step Solution:
Step 1: Compute radius r from the diameter: r = 42 / 2 = 21 cm. Step 2: Use formula TSA = 3 * pi * r^2. Step 3: Substitute pi = 22 / 7 and r = 21 cm. Step 4: Compute r^2: 21^2 = 441. Step 5: So TSA = 3 * (22 / 7) * 441. Step 6: Simplify 441 / 7 = 63. Step 7: TSA = 3 * 22 * 63. Step 8: First compute 22 * 63 = 1386. Step 9: Then TSA = 3 * 1386 = 4158 square centimetres.

Verification / Alternative check:
Alternatively, find curved surface area and base area separately. CSA = 2 * pi * r^2 = 2 * (22 / 7) * 441 = 2 * 22 * 63 = 2772 square centimetres. Base area = pi * r^2 = (22 / 7) * 441 = 22 * 63 = 1386 square centimetres. Summing: 2772 + 1386 = 4158 square centimetres, matching the previous calculation.

Why Other Options Are Wrong:
3500 square centimetres: Underestimates the total area; likely due to dropping part of CSA or base area.
5782 and 6321 square centimetres: These overestimate the area and do not match any correct combination of CSA and base area with radius 21 cm.
7782 square centimetres: Far too large and inconsistent with the radius given.

Common Pitfalls:
A very common mistake is to use only the curved surface area formula 2 * pi * r^2 and forget to add the area of the base. Another error is to incorrectly halve or double the sphere surface area instead of applying the specific hemisphere formulas. Always remember that for a solid hemisphere you must include the base to get the total surface area.

Final Answer:
The total surface area of the hemisphere is 4158 square centimetres.