Question

Perspective view of two identical hollow cylinders is shown below. Assume the outer diameter and length of each cylinder are equal. The two cylinders intersect each other at 90 degrees to form a union, such that their centroids (centre of axis) coincide. How many visible surfaces will the resultant union have?

• A. 24
• B. 32
• C. 28
• D. 20

Hint:

For intersecting hollow shapes, first count the untouched flat end caps, then use symmetry to break down the outer curved walls and inner passages into simple quadrants.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question belongs to the topic of

3D Geometry and Boolean Solid Modeling.
We are asked to calculate the total number of distinct visible surfaces (both flat and curved) formed by the 3D union of two identical hollow cylinders intersecting orthogonally at their midpoints.

Step 2: Key Formula or Approach:
We count the surfaces of the individual cylinders and then account for how those surfaces are divided or merged by the 30-degree intersection:

• A single hollow cylinder has 4 surfaces: 1 outer curved surface, 1 inner curved surface, and 2 flat annular end rings.

• When they intersect, the union merges parts of the outer walls and splits the internal passages.


Step 3: Detailed Explanation:

• Let us perform a systematic surface count on the union:
- Annular End Faces: The 4 circular ends of the two cylinders do not overlap or intersect because the intersection is concentrated at their central midpoints. This contributes exactly 4 flat annular surfaces.
- Outer Curved Surfaces: The outer walls of the two cylinders intersect along a 3D saddle curve. This intersection divides the outer curved surface of each cylinder into 4 separate quadrants. In the union, these quadrants merge to form 4 distinct curved outer surfaces.
- Inner Curved Surfaces: The inner walls of the hollow cylinders intersect in a similar manner. The interior chamber is divided into 12 curved inner segments where the perpendicular passages meet.

• Adding these components together:
\[ \text{Total Surfaces} = 4 \text{ (ends)} + 4 \text{ (outer)} + 12 \text{ (inner)} = 20 \text{ surfaces} \]


Step 4: Final Answer:
The union has exactly 20 visible surfaces, corresponding to Option (D).