A solid cube has side length 8 centimetres. It is cut by two planes along both diagonals of its top face, producing four equal congruent solid parts. What is the total surface area, in square centimetres (cm^2), of each of these four parts including all faces that become exposed after cutting?

Introduction / Context:
This solid geometry question studies how cutting a cube affects the surface area of the resulting pieces. A cube of side 8 centimetres is sliced along both diagonals of its top face by vertical planes, creating four congruent wedge shaped solids. We are asked for the total surface area of one such part, counting all faces, including the newly created cut surfaces, which is important in practical slicing or machining problems.

Given Data / Assumptions:

• Original solid is a cube with side length a = 8 cm.
• Two vertical planes pass through the diagonals of the top face and extend through the cube.
• These cuts divide the cube into four equal congruent solids.
• Total surface area of one part must include original outer faces plus newly exposed cut faces.
• All edges of the cube remain straight and exact; no material is lost in cutting.

Concept / Approach:
Instead of trying to visualise every face of one part directly, a more efficient approach is to look at the total surface area of all four parts together. Initially, the cube has a known surface area. When we cut along a plane inside the cube, each new cutting face is shared by two parts. Therefore, the total surface area of all four parts equals the original outer surface area plus twice the total area of all cutting planes. Once that combined area is known, we divide by four to get the surface area of each congruent part.

Step-by-Step Solution:
Step 1: Surface area of the original cube is S_cube = 6 * a^2 = 6 * 8^2 = 6 * 64 = 384 cm^2. Step 2: Each diagonal of the top face has length a√2 = 8√2 cm. Step 3: Each cutting plane is vertical and has a rectangular intersection inside the cube of size 8 (height) by 8√2 (width), so area of each plane = 8 * 8√2 = 64√2 cm^2. Step 4: There are two such cutting planes, so total internal cutting area = 2 * 64√2 = 128√2 cm^2. Step 5: When counting the surface areas of all four parts, each cutting surface appears twice (once on each of the two touching solids), so the extra area added is 2 * 128√2 = 256√2 cm^2. Step 6: Total surface area of all four parts together = original outer area + extra area = 384 + 256√2 cm^2. Step 7: Since all four parts are congruent, surface area of each part = (384 + 256√2) / 4 = 96 + 64√2 cm^2.

Verification / Alternative check:
An alternate viewpoint is to observe that the top and bottom faces of each part are right angled isosceles triangles, and the side faces are rectangles and right angled quadrilaterals. Computing each face separately will also lead to the same total of 96 + 64√2 cm^2 per piece, but this is more time consuming. Using symmetry and counting cutting planes is therefore a more elegant and exam friendly method, and it reproduces the same final expression.

Why Other Options Are Wrong:
80 + 64√2 and 96 + 48√2 both arise from partially counting either the original faces or the cutting faces. In one case, the contribution of the original cube faces is underestimated; in the other, the doubled effect of cuts on two solids is ignored. 80 + 48√2 and 112 + 64√2 are further from the correct total and do not satisfy the requirement that four times the individual surface area minus the original cube area equals twice the total cutting plane area.

Common Pitfalls:
Many students try to visualise every face of one part directly and often forget some faces or double count others. It is also easy to mistake the area of the cutting rectangles by using the side 8 instead of the diagonal 8√2. Another issue is neglecting that each cut face appears on two solids, which changes the total contribution when summing areas. Using a systematic global approach, starting from the whole cube and then dividing by four, greatly reduces such errors.

Final Answer:
The total surface area of each of the four parts is 96 + 64√2 cm^2.