AMC 10 · 2008 · #21
Grade 8 geometry-3dA cube with side length 1 is sliced by a plane that passes through two diagonally opposite vertices A and C and the midpoints B and D of two opposite edges not containing A or C, as shown. What is the area of quadrilateral ABCD?
Pick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: The cube has side length 1.; A and C are diagonally opposite vertices of the cube (a space diagonal apart).; B and D are midpoints of two opposite edges, neither edge touching A or C.; ABCD is the flat cross-section where the plane meets the cube.
Unknowns: The area of quadrilateral ABCD.
Understand
Restated: 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.
Givens: The cube has side length 1.; A and C are diagonally opposite vertices of the cube (a space diagonal apart).; B and D are midpoints of two opposite edges, neither edge touching A or C.; ABCD is the flat cross-section where the plane meets the cube.
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #4 Introduce a Variable, #1 Draw a Diagram, #7 Identify Subproblems
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.
Execute — Answer: A
6.NS.C.6 Step 1 Put the cube on coordinates
- Set the cube so its corners are all the points whose x, y, z are each 0 or 1.
- Take A and C as the opposite corners A = (1,0,0) and C = (0,1,1); they differ in every coordinate, which is exactly what 'diagonally opposite' means.
- The two opposite edges that miss both A and C run from (0,0,0) to (0,0,1) and from (1,1,0) to (1,1,1); their midpoints are B = (0,0,1/2) and D = (1,1,1/2).
- Now every point of ABCD is a concrete triple.
💡 Giving each corner an address turns a shape you have to imagine into numbers you can just compute with.
8.G.B.8 Step 2 Show ABCD is a rhombus
- Measure the four sides with the 3D distance formula (Pythagoras with three legs).
- Going A to B changes x by 1, y by 0, z by 1/2, so AB = sqrt(1 + 0 + 1/4) = sqrt(5)/2.
- Doing the same for BC, CD, DA gives sqrt(5)/2 every time.
- Four equal sides means ABCD is a rhombus.
💡 Equal-length sides are the fingerprint of a rhombus, and the distance formula lets you check all four at once.
8.G.B.7 Step 3 Measure the two diagonals
- The diagonals of ABCD are AC and BD.
- AC joins two opposite cube corners, a space diagonal: AC = sqrt(1 + 1 + 1) = sqrt(3).
- BD joins B = (0,0,1/2) and D = (1,1,1/2), which changes x by 1, y by 1, z by 0, so BD = sqrt(1 + 1 + 0) = sqrt(2), the length of a face diagonal.
- (As a rhombus should, these diagonals cross at the shared midpoint (1/2,1/2,1/2) at a right angle.)
💡 The long diagonal cuts straight through the cube while the short one lies flat on a face, and both are just Pythagoras in space.
7.G.B.6 Step 4 Apply the rhombus area formula
- A rhombus's area is half the product of its diagonals, because its two diagonals are perpendicular and split it into four matching right triangles.
- So the area is (1/2) times sqrt(3) times sqrt(2).
- Multiplying the roots, sqrt(3)*sqrt(2) = sqrt(6), so the area is sqrt(6)/2.
- That is choice (A).
💡 Perpendicular diagonals let you build the rhombus from four right triangles, which is exactly the half-product-of-diagonals shortcut.
6.NS.C.6 Set the cube so its corners are all the points whose x, y, z are each 0 or 1. Ta 8.G.B.8 Measure the four sides with the 3D distance formula (Pythagoras with three legs) 8.G.B.7 The diagonals of ABCD are AC and BD. AC joins two opposite cube corners, a space 7.G.B.6 A rhombus's area is half the product of its diagonals, because its two diagonals Review
Reasonableness: 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 beats 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 5/8 = 0.625 or 3/4 = 0.75 are smaller than one face and can be ruled out immediately.
Alternative: Skip coordinates and use the two diagonals directly from cube facts: any space diagonal of a unit cube is sqrt(3) and any face diagonal is sqrt(2). By symmetry the cross-section is a rhombus with exactly these two diagonals, so its area is (1/2)(sqrt(3))(sqrt(2)) = sqrt(6)/2. Same answer with almost no computation.
CCSS standards used (min grade 8)
6.NS.C.6Understand a rational number as a point on the number line (Assigning coordinate addresses to the cube's vertices and edge midpoints so the cross-section becomes numeric.)8.G.B.8Apply the Pythagorean theorem to find distance between two points in a coordinate system (Computing all four side lengths as sqrt(5)/2 to prove ABCD is a rhombus.)8.G.B.7Apply the Pythagorean theorem to determine unknown side lengths in right triangles (Finding the space-diagonal length sqrt(3) and face-diagonal length sqrt(2) of the rhombus.)7.G.B.6Solve real-world problems involving area, surface area, and volume (Using the half-product-of-diagonals area formula to get sqrt(6)/2.)
⭐ 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.
⭐ 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.
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