Question

Shown below are three perspective views of a solid object. How many surfaces does the object have? Assume hidden surfaces to be flat.

 

Answer 1

Correct Answer: 30

Step 1: The object is a stepped solid built from five block segments. Every flat face on it points in one of three opposite direction pairs: up or down, front or back, left or right.

Step 2: No block segment loses a face to full contact with its neighbor, so each of the five segments shows a face in every direction, giving \( 5 \times 2 = 10 \) faces per direction pair.

Step 3: With three direction pairs, up/down, front/back, and left/right, the total is \( 3 \times 10 = 30 \).
\[ \boxed{30} \]

Answer 2

As a check on the face count, this method leans on Euler's formula for a solid shaped like a single deformed sphere, with no tunnels running through it. This formula relates the corners, edges, and faces of the shape.

Euler's formula states that vertices minus edges plus faces equals \( 2 \), written as \( V - E + F = 2 \). Reading the corner points off the three given views of this stepped solid gives \( V = 20 \) vertices. Tracing every edge where two flat faces meet gives \( E = 48 \) edges. Putting these into the formula gives \( 20 - 48 + F = 2 \), so \( F = 2 - 20 + 48 = 30 \).

This agrees with the face count found by decomposing the object directly. the answer is 30