AMC 10 · 2008 · #21

Grade 8 geometry-3d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

coordinate-geometrypythagorean-theoremspatial-visualization spatial-visualizationconvert-to-algebra ↑ Prerequisites: coordinate-geometry

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A unit cube is cut by a flat plane. The plane goes through two opposite corners A and C of the cube and through the midpoints B and D of two opposite edges that touch neither A nor C. The cut leaves a four-sided figure ABCD on the plane. Find its area.

Pick an answer.

(A)

$\ \frac{\sqrt{6}}{2}$

(B)

$\ \frac{5}{4}$

(C)

$\ \sqrt{2}$

(D)

$\ \frac{5}{8}$

(E)

$\ \frac{3}{4}$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Visualize Spatial Relationships

A slanted slice through a 3D cube is hard to measure by eye, so pin every point to coordinates on the cube. Once the four corners are numbers, the distance formula turns the shape question into arithmetic: check the side lengths to name the figure, then measure its diagonals. The figure turns out to be a rhombus, and a rhombus has a clean area shortcut, half the product of its diagonals, so the whole problem collapses to finding two diagonal lengths.

1STEP 1

Put the cube on coordinates

Put every corner at coordinates 0 or 1, so the section's points are $A=(1,0,0)$ , $B=(0,0,\frac{1}{2})$ , $C=(0,1,1)$ , $D=(1,1,\frac{1}{2})$ .

A=(1,0,0), B=(0,0,1/2), C=(0,1,1), D=(1,1,1/2)

2STEP 2

Show ABCD is a rhombus

The 3D distance formula gives all four sides as $\frac{\sqrt{5}}{2}$ , so ABCD is a rhombus.

AB=√(1²+0²+(1/2)²)=√5/2=BC=CD=DA

3STEP 3

Measure the two diagonals

Its diagonals are the cube's space diagonal $AC=\sqrt{3}$ and a face diagonal $BD=\sqrt{2}$ , meeting at right angles.

AC=√(1²+1²+1²)=√3, BD=√(1²+1²+0²)=√2

4STEP 4

Apply the rhombus area formula

Half the product of the perpendicular diagonals, $\frac{1}{2}\cdot\sqrt{3}\cdot\sqrt{2}$ , gives $\frac{\sqrt{6}}{2}$ — choice (A).

[ABCD]=1/2 d₁ d₂=1/2·√3·√2=√6/2

Answer

√(6)/2

\frac{\sqrt{6}}{2}

is about 1.22. A full face of the cube has area 1, and this slanted slice should be a bit bigger than a face since it stretches from corner to corner, so a value just above 1 is exactly right. It also stays under a diagonal rectangle of size 1 by

\sqrt{2}

(area 1.41 would be the widest reasonable slice), so 1.22 sits sensibly between a face and that widest cut. Choices like

\frac{5}{8}

= 0.625 or

\frac{3}{4}

= 0.75 are smaller than one face and can be ruled out immediately.

💡Key takeaway

Give a 3D cube corner-and-midpoint coordinates, and a slanted slice becomes a rhombus whose area is just half of its two diagonals multiplied together.

Put the cube on coordinates
Show ABCD is a rhombus
Measure the two diagonals
Apply the rhombus area formula

Put the cube on coordinates

Show ABCD is a rhombus

Measure the two diagonals

Apply the rhombus area formula

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