Question

Two perspective views of the same solid object are shown. Count the total number of surfaces in the object. Assume hidden surfaces are flat.

Hint:

For “count the faces” problems: (1) partition the object into simple features, (2) count faces per feature, (3) subtract shared/continuous planes once to avoid double-counting.

Answer 1

Step 1: Break the solid into feature-groups.
From the two views we can identify:
(i) a thick ring-like frame with a heart-shaped through–hole,
(ii) a vertical spine on the right,
(iii) three tooth-like steps at the top,
(iv) three tooth-like steps at the bottom,
(v) a recessed inner ledge following the heart contour.
Step 2: Count large planar faces.
- Left outer side face and right outer side face of the thick frame: \(2\).
- Front and back faces of the vertical spine (it merges smoothly with the frame and contributes its own planar walls): \(2\).
- The inner heart through–hole produces two side faces (one on each thickness side): \(2\).
Subtotal so far: \(6\).
Step 3: Count faces created by the stepped “teeth”.
Each tooth contributes: a top, a bottom, a front riser, a rear riser and two lateral faces (left and right), i.e. \(6\) faces per tooth.
- Top set: \(3 \times 6 = 18\).
- Bottom set: \(3 \times 6 = 18\).
But the outermost top and bottom horizontal planes are continuous with the frame; for the three–tooth staircase, the continuous outer planes account for one shared top and one shared bottom across the row, effectively reducing by \(2\) duplicates for each row.
Thus effective for each row: \(18 - 2 = 16\).
Two rows (top and bottom): \(32\).
Step 4: Faces from the inner recessed ledge.
The inset ledge that traces the heart contour adds one continuous vertical wall and one continuous horizontal shelf on each side of the thickness: \(4\) faces.
Step 5: Sum up.
Large faces \(6\) + stepped regions \(32\) + ledge faces \(4\) \(=\) \(42\) faces. Final Answer: \[ \boxed{42 \text{ surfaces}} \]