Which solid corresponds to the given top view, front view, and side view?

Solids to choose from:

Question

Which solid corresponds to the given top view, front view, and side view? Solids to choose from:

Homi · Accepted Answer

Understand the Question

A 3D solid composed of cubes can be uniquely identified by observing it from three orthogonal perspectives:
- Front View: The 2D projection seen directly from the front.
- Top View: The 2D projection seen directly from above.
- Side View: The 2D projection seen from the side.
By analyzing the number and position of blocks in each given 2D view and comparing them systematically to solids (i) through (vii), we find the single solid that matches all three views simultaneously.

Step 1 · Analyze the Given 2D Views

Question diagram

Front View ( $X\text{-}Z$ plane):
- A vertical column of $3$ blocks on the left at $x = 0$ .
- A horizontal arm of $2$ blocks extending to the right at the middle height ( $z = 1$ ) at $x = 1$ and $x = 2$ .
- Dimensions: $\text{width} = 3\text{ units}, \text{height} = 3\text{ units}$ .
Top View ( $X\text{-}Y$ plane):
- A horizontal row of $3$ blocks at the back ( $y = 0$ , across $x = 0, 1, 2$ ).
- A vertical column of $2$ blocks extending forward at $x = 0$ ( $y = 1$ ).
- Dimensions: $\text{width} = 3\text{ units}, \text{depth} = 2\text{ units}$ .
Side View ( $Y\text{-}Z$ plane):
- A vertical column of $3$ blocks at the front ( $y = 0$ , across $z = 0, 1, 2$ ).
- At the back ( $y = 1$ ), one block at the top ( $z = 2$ ) and one block at the bottom ( $z = 0$ ), with the middle block ( $z = 1$ ) missing.
- Dimensions: $\text{depth} = 2\text{ units}, \text{height} = 3\text{ units}$ .

Step 2 · Evaluate Solids (i) to (vii)

Compare each candidate solid with the three given views:

Solid (i): Front and Top views match, but its Side view forms a solid $2 \times 3$ rectangle with no middle gap (incorrect).
Solid (ii): The horizontal arm is at the bottom instead of the middle level in the Front view (incorrect).
Solid (iii): Top view has only $1$ block at the front-left and $3$ blocks at the back (incorrect).
Solid (iv): The horizontal arm is at the top instead of the middle level in the Front view (incorrect).
Solid (v):
- Front View: $3$ -block vertical column on the left with a $2$ -block middle arm (matches).
- Top View: $3$ -block row at the back with a column extending forward (matches).
- Side View: $3$ -block vertical column with top and bottom blocks at the back and a gap in the middle (matches).
Solid (vi): Top view does not match the given configuration (incorrect).
Solid (vii): Top view does not match the given configuration (incorrect).

Therefore, Solid (v) is the only solid that corresponds to all three views.

Answer

(v)

Common Mistakes

Overlooking View Gaps: Missing the empty middle space at the back in the side view, which leads to incorrectly choosing solid (i).
Confusing Levels in Front View: Misidentifying which vertical level the horizontal arm extends from (middle vs. top/bottom in solids ii and iv).
Swapping Top and Front Orientations: Mixing up depth ( $y$ -axis) and height ( $z$ -axis) when projecting 3D blocks onto 2D views.

Question 15