A cube 4 cm x 4 cm x 4 cm has all its corners chamfered as shown in the figure below. On each of its faces it has got a small tetrahedral notch having edges of 1 cm each. What is the sum of the number of vertices and the number of edges in this solid?
Show Hint
When dealing with complex 3D shapes, break down the problem. Calculate the properties of the base shape first, then calculate the net change (vertices, edges, faces added or removed) for each modification separately. Summing these up gives the final count.
Step 1: Understanding the Question:
We need to find the total number of vertices (corners) and edges of a complex solid created by modifying a standard cube in two ways: chamfering all corners and adding a tetrahedral notch on each face. Step 2: Detailed Explanation:
Let's analyze the process by starting with a simple cube and applying the modifications one by one. Part A: The Standard Cube
A standard cube has:
- Vertices (V) = 8
- Edges (E) = 12 Part B: Chamfering the 8 Corners
Chamfering a corner means slicing it off. This operation replaces each original vertex with a new triangular face.
- Vertices: Each of the 8 original vertices is replaced by 3 new vertices (the corners of the new triangle).
New number of vertices = \(8 \text{ corners} \times 3 \text{ vertices/corner} = 24\).
- Edges: The 12 original edges of the cube remain (though they are shortened). Each of the 8 chamfering operations creates 3 new edges that form the new triangular face.
New number of edges = \(12 \text{ (original)} + (8 \text{ corners} \times 3 \text{ new edges/corner}) = 12 + 24 = 36\).
So, after chamfering, the solid has V = 24 and E = 36. Part C: Adding 6 Tetrahedral Notches
A tetrahedral notch is created on each of the 6 original faces. The notches are separate from the chamfered corners. A tetrahedron has 4 vertices and 6 edges. Adding a notch means creating this geometry on the face.
- Vertices Added: For each notch, we add 4 new vertices (3 forming a triangle on the face, and 1 apex point inside the cube). These vertices are new and do not overlap with the 24 vertices from the chamfered corners.
Total vertices added = \(6 \text{ faces} \times 4 \text{ vertices/notch} = 24\).
- Edges Added: For each notch, we add 6 new edges (3 forming the triangular base on the face, and 3 connecting the base to the inner apex).
Total edges added = \(6 \text{ faces} \times 6 \text{ edges/notch} = 36\). Part D: Final Calculation
Now we combine the results from the chamfered cube and the added notches.
- Total Vertices = (Vertices of chamfered cube) + (Vertices added by notches)
Total V = 24 + 24 = 48.
- Total Edges = (Edges of chamfered cube) + (Edges added by notches)
Total E = 36 + 36 = 72. Step 3: Final Answer:
The question asks for the sum of the number of vertices and the number of edges.
\[ \text{Sum} = \text{Total Vertices} + \text{Total Edges} = 48 + 72 = 120 \]
The sum is 120.
