A solid sphere of radius 21 centimetres is cut by three mutually perpendicular planes through its centre, one along each coordinate axis, so that it is divided into 8 identical parts. What is the total surface area of each part, in square centimetres?

Introduction / Context:
This question involves a sphere that is divided into eight equal parts by three mutually perpendicular planes through its centre. Each part is often called an octant of the sphere. The surface area of each part includes a portion of the original curved surface and several flat circular pieces created by the cuts. Understanding how these areas are distributed is key to finding the total surface area of each part.

Given Data / Assumptions:
- Radius of the sphere r = 21 cm.
- The sphere is cut by three planes through its centre, each perpendicular to the others, similar to the coordinate planes x = 0, y = 0 and z = 0.
- These cuts divide the sphere into 8 congruent parts or octants.
- We need the total surface area of one such part, including both curved and planar surfaces.

Concept / Approach:
The total surface area of the full sphere is:
S sphere = 4 * π * r^2
When the sphere is divided into 8 octants by three perpendicular planes through the centre, each octant has:
- One eighth of the original curved surface area of the sphere.
- Three flat faces, each coming from one of the circular cross sections created by the planes. Each plane cuts the sphere along a great circle of radius r, whose area is π * r^2. For a single octant, each of these three disks contributes one quarter of its area, because each disk is divided into four equal sectors by the other two planes.
So for each octant, total area = (curved area for that octant) + (area of three quarter disk sectors).

Step-by-Step Solution:
Step 1: Compute total surface area of the sphere: S sphere = 4 * π * r^2. Step 2: With r = 21 cm and π taken as 22 / 7, S sphere = 4 * 22 / 7 * 21^2 = 4 * 22 / 7 * 441. Step 3: Simplify 441 / 7 = 63, so S sphere = 4 * 22 * 63 = 88 * 63 = 5544 square centimetres. Step 4: The curved surface area for one octant is one eighth of this: 5544 / 8 = 693 square centimetres. Step 5: Each great circle formed by a cutting plane has area π * r^2 = 22 / 7 * 21^2 = 22 / 7 * 441 = 22 * 63 = 1386 square centimetres. Each such disk is divided into four equal sectors by the other two planes, so one octant includes one quarter of each disk. Step 6: The area from one disk in a single octant is 1386 / 4 = 346.5 square centimetres. Since each octant touches three such disks, total planar area per octant is 3 * 346.5 = 1039.5 square centimetres. Step 7: Add the curved and planar areas: total area per part = 693 + 1039.5 = 1732.5 square centimetres.

Verification / Alternative check:
An alternative perspective is to note that the total new area created by the cuts across all octants must equal the combined areas of the three great circle disks, each of area 1386. Each disk is cut into four sectors and each octant contains three of these sectors. Accounting for all eight octants, you recover exactly the three full disks. Combining this with the fact that the curved area is evenly distributed among the eight octants again leads to the same total area per octant.

Why Other Options Are Wrong:
844.5 and 1039.5 represent partial contributions from either the curved portion or the planar faces alone, not both together.
1115.6 is not consistent with any clean fraction of the known areas.
1386 is the area of one complete great circle and does not represent the total area of an octant.

Common Pitfalls:
Learners often forget to include the flat faces introduced by the cutting planes and only count the curved surface, or they miscount how many sectors from each disk belong to one octant. Another error is to assume that only one disk contributes to each part, instead of three. Carefully visualising how the planes partition the sphere helps avoid these mistakes.

Final Answer:
The total surface area of each part is 1732.5 square centimetres.