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Area of an oblique section of a rectangular prism

Find the section shape before its area

A plane through one edge of a rectangular prism cuts the opposite parallel face along a parallel segment. Identify the resulting polygon, then find any sloping side from a right triangle on a side face.

Problem

In a rectangular prism $A B C D A_{1} B_{1} C_{1} D_{1}$ , let $A B = 4$ , $A D = 25$ and $A A_{1} = 4$ . Point $K$ is the midpoint of $B B_{1}$ . Find the area of the section through $A_{1}$ , $D_{1}$ and $K$ .

Answer

20

square units.

The section is a square with side 2√5, though it is tilted relative to the horizontal faces.

Step-by-step solution

1. Identify the fourth corner

The plane meets the side face $B C C_{1} B_{1}$ along a segment $K M$ parallel to $A_{1} D_{1}$ , where $M$ is the midpoint of $C C_{1}$ . The section is the rectangle $A_{1} D_{1} M K$ . Its two directions are perpendicular because $A D$ is perpendicular to the face $A B B_{1} A_{1}$ .

2. Find the two side lengths

One side is $A_{1} D_{1} = A D = 25$ . On face $A B B_{1} A_{1}$ , the horizontal difference from $A_{1}$ to $K$ is $A B = 4$ and the vertical difference is half the height, $2$ . Thus

A_{1} K = 4^{2} + 2^{2} = 20 = 25 .

3. Multiply

S = A_{1} D_{1} \cdot A_{1} K = (25)^{2} = 20.

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