The figure below represents the plan and elevation of a square based prism of base 4 cm x 4 cm and height 20 cm. The prism is standing on the horizontal plane (HP) with its faces making an angle of 45 degrees with the vertical plane (VP). A cutting plane, perpendicular to VP, divides the prism as shown in the figure below.
The surface area (in cm\(^2\)) of the vertical faces of the prism below the cutting plane is (in integer).
Show Hint
Read the cutting heights at edges 1 and 3 from the elevation (6 cm and 12 cm), average them for the middle edge (9 cm), then sum the four trapezoidal face areas (each width 4 cm).
Step 1: Use a shortcut for the total lateral area below a sloped cut.
For a prism whose vertical edges are cut at different heights by a plane, the total area of all side faces below the cut equals the edge width (here, the base side length) multiplied by the SUM of the cutting heights at all the vertical edges. This works because each face's trapezoidal area uses the average of its two edge heights, and going all the way around the prism, every edge height gets counted exactly twice (once for each face meeting at that edge), so the halves cancel out neatly into a single sum.
Step 2: Find the four cutting heights.
As read from the figure, the sloping cutting-plane line meets edge 1 at $6$ cm and edge 3 at $12$ cm (the two outer edges in the elevation). Edge $(2,4)$ sits exactly midway between them in the elevation, so its height is the average:
$$h_{2,4} = \frac{6+12}{2} = 9 \text{ cm}$$
So the four heights, going around the prism, are $6, 9, 12, 9$ cm (edges 1, 2, 3, 4 respectively).
Step 3: Add up the four heights.
$$6+9+12+9 = 36 \text{ cm}$$
Step 4: Multiply by the base edge length (4 cm).
$$\text{Total lateral area below the cut} = 4 \times 36 = 144 \text{ cm}^2$$
Step 5: Check against the face-by-face method.
Working out each trapezoidal face separately gives $30, 42, 42, 30$ cm$^2$, which also sums to $144$ cm$^2$, confirming the shortcut.
Final Answer:
The surface area of the vertical faces of the prism below the cutting plane is 144 cm$^2$.
