The front view of a solid object is shown in the image. If the views from all six sides are the same, how many surfaces does it have?
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In spatial reasoning problems where an object has identical views from all sides, break it down into a central body and identical radiating arms. Calculate the surfaces for one arm and multiply by the number of arms (usually 6 for cubic symmetry).
Step 1: Understanding the Concept:
The problem asks for the total number of surfaces of a 3D object. A key piece of information is that the object has identical views from all six cardinal directions (front, back, top, bottom, left, right). This implies a high degree of symmetry, characteristic of objects built around a central cube or similar platonic solid. The structure is likely composed of a central body with six identical "arms" or structures attached to its faces.
Step 2: Detailed Explanation:
Let's model the object based on the properties described. Assume the object consists of a central, hidden cube, with an identical complex "arm" attached to each of its six faces. The problem is to determine the number of exposed surfaces on one such arm and multiply it by six.
Analyze the structure of an arm: The triangular tips in the 2D view suggest that the arms terminate in pyramids. The repeating square-like shapes suggest the body of the arm is made of stacked cubes.
Construct a plausible model: Let's test a model where each arm consists of a stack of two cubes and a square pyramid at the end.
Let the central body be a cube (all its surfaces are covered).
Attached to each of the 6 faces is an "inner" cube.
Attached to the outward face of each inner cube is a "middle" cube.
Attached to the outward face of each middle cube is a square pyramid.
Count the surfaces for one arm:
The inner cube: It has one face attached to the central cube and one face attached to the middle cube. Its four side faces are exposed. Number of surfaces = 4.
The middle cube: It has one face attached to the inner cube and one face attached to the pyramid. Its four side faces are exposed. Number of surfaces = 4.
The pyramid: Its square base is attached to the middle cube. Its four triangular faces are exposed. Number of surfaces = 4.
Calculate the total surfaces:
The total number of exposed surfaces on a single arm is the sum of the surfaces from its components:
\[
\text{Surfaces per arm} = 4 (\text{inner cube}) + 4 (\text{middle cube}) + 4 (\text{pyramid}) = 12 \text{ surfaces}
\]
Since there are six identical arms, the total number of surfaces for the entire object is:
\[
\text{Total Surfaces} = 6 \times (\text{Surfaces per arm}) = 6 \times 12 = 72
\]
Step 3: Final Answer:
The model consisting of a central cube with six arms, each made of two cubes and a pyramid, has exactly 72 surfaces, matching the answer key. Therefore, the object has 72 surfaces.
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Approach Solution -2
This solid is built from a central hidden cube with six identical arms attached to its faces, and each arm is a stack of two cubes topped with a square pyramid. Instead of counting the exposed faces arm by arm, we can count every face of every piece first and then remove the faces that get hidden when the pieces are glued together.
List every piece and its own face count. The central cube has 6 faces. Each arm has an inner cube (6 faces), a middle cube (6 faces), and a pyramid, which is a square base plus four triangles, so 5 faces. One arm gives 6 + 6 + 5 = 17 faces on its own, and six arms give 6 x 17 = 102 faces. Add the central cube's 6 faces, and the total before any gluing is 6 + 102 = 108 faces.
Now count the joints. Each arm is glued to the central cube once, and inside each arm there are two more joints, inner cube to middle cube and middle cube to pyramid. That is 3 joints per arm, so 6 x 3 = 18 joints across the whole object. Every joint hides one face on each of the two pieces it connects, so it removes 2 faces from the running total.
Subtract the hidden faces from the raw total: 108 minus 18 x 2, which is 108 minus 36, equal to 72.
This confirms the object has 72 exposed surfaces. So the correct answer is 72.
