The figure shows perspective views of the same solid object. Which option shows the count of the total number of surfaces?
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When counting surfaces on a complex 3D object, break it down into simpler geometric components or categorized surfaces (e.g., top, bottom, sides, features). Mentally "paint" each surface to keep track of what you've counted.
Step 1: Sort every surface by which way it faces.
Instead of grouping surfaces by which part of the object they belong to, group them by direction, upward-facing, downward-facing, or vertical, side-facing.
Step 2: Count the upward-facing surfaces.
These are the main top surface, the top of the raised block, the top of the cylindrical part, and the curved floor of the groove, giving 4 upward-facing surfaces.
Step 3: Count the downward-facing surfaces.
These are the single flat surface the whole object rests on and the underside of the cylinder's overhang, giving 2 downward-facing surfaces.
Step 4: Count the vertical, side-facing surfaces.
These include the two groove walls, the cylinder's curved side and the vertical face it overhangs, the block's front and right faces, and the main body's front, back, left, right and inner corner faces, giving 11 vertical surfaces.
Step 5: Add the three groups together.
\[ 4 + 2 + 11 = 17 \]
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Approach Solution -2
A third way to count is to treat the object as several simple solids glued together, count each solid's full surface separately, and then subtract the faces that get hidden where the pieces join.
Main L-shaped block: a simple block like this has 6 outer faces, but with the groove cut into its top, two more wall faces and one floor face are added, and the top face is split by the groove without changing the total count, giving 9 faces for the main body plus groove.
Raised rectangular block: on its own this smaller block has 6 faces, but the face resting against the main body is hidden, leaving 5 visible faces contributed to the whole object.
Cylindrical part: a cylinder normally has 3 surfaces, its top, its curved side, and its bottom, but since it overhangs part of the main body, its bottom is replaced by the visible underside plus the vertical face of the main body it covers, contributing 3 visible surfaces of its own.
Adding the visible surfaces from each piece, 9 from the main body and groove, 5 from the raised block, and 3 from the cylinder, gives a total of 17 distinct surfaces once the hidden joining faces are left out.
