AMC 10 · 2011 · #24
Grade 8 geometry-3dTwo distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
Pick an answer.
AMC 10 2011 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 has 8 corners. Two different regular tetrahedra are built using only these corners as their vertices. Find the volume of the solid region that lies inside both tetrahedra at once.
Givens: A cube with edge length 1.; Two distinct regular tetrahedra, each using 4 of the cube's 8 vertices.; Every edge of each tetrahedron is a diagonal of a face of the cube.
Unknowns: The volume of the overlap (intersection) of the two tetrahedra.
Understand
Restated: A unit cube has 8 corners. Two different regular tetrahedra are built using only these corners as their vertices. Find the volume of the solid region that lies inside both tetrahedra at once.
Givens: A cube with edge length 1.; Two distinct regular tetrahedra, each using 4 of the cube's 8 vertices.; Every edge of each tetrahedron is a diagonal of a face of the cube.
Plan
Primary tool: #17 Visualize Spatial Relationships
Secondary: #1 Draw a Diagram, #7 Identify Subproblems
This is a pure 3D picture problem, so the main move is to see the two tetrahedra sitting in the cube and picture where they overlap. Placing the cube on coordinates pins down every vertex. Seeing that the shared region is an octahedron whose corners are the cube's face centers is the key insight, and then the volume becomes an easier subproblem: cut the octahedron into two simple square pyramids.
Execute — Answer: D
5.G.A.2 Step 1 Pick the two tetrahedra by alternating corners
- Put the cube at coordinates from (0,0,0) to (1,1,1).
- Color the 8 corners like a checkerboard: a corner is one color if its coordinates add to an even number, the other color if odd.
- The 4 even corners form one regular tetrahedron and the 4 odd corners form the other.
- The two tetrahedra use complementary sets of corners, so together they fill all 8 vertices of the cube.
💡 Alternating corners of a cube is the only way to get four mutually equidistant points, which is what a regular tetrahedron needs.
8.G.B.7 Step 2 Each edge is a face diagonal of length root two
- Take two corners of one tetrahedron, such as (0,0,0) and (1,1,0).
- They differ by 1 in two coordinates and by 0 in the third, so the distance between them is the diagonal of a unit square face.
- By the Pythagorean theorem this length is the square root of 1 squared plus 1 squared, which is root two.
- Every edge of both tetrahedra is such a face diagonal, so all six edges are equal and each tetrahedron really is regular.
💡 Equal edges of a solid must all be the same kind of segment, and here they are all face diagonals.
7.G.A.3 Step 3 See the overlap as an octahedron at the face centers
- Each face of the cube has two diagonals: one is an edge of Tetra 1 and the other is an edge of Tetra 2, and they cross at the center of that face.
- The region inside both tetrahedra is the block around the cube's center bounded by these crossings, and its six corners are exactly the six face centers of the cube.
- Six corners with a square all the way around the middle and a point above and below is an octahedron.
💡 The two face diagonals of every face cross at its center, so those centers are the natural corners of the shared region.
6.G.A.1 Step 4 Find the square through the middle of the octahedron
- Slice the octahedron by the horizontal plane halfway up the cube, at height one half.
- The four face centers on the four side faces all lie in this plane and form a square.
- Its two diagonals run straight across the cube, from (0, 1/2, 1/2) to (1, 1/2, 1/2) and from (1/2, 0, 1/2) to (1/2, 1, 1/2), so each diagonal has length 1.
- The area of a square is half the product of its diagonals, giving one half.
💡 The two long diagonals of the octahedron are full edges of the cube laid across the middle, so each is length 1.
7.G.B.6 Step 5 Add the two square pyramids
- That middle square splits the octahedron into two square pyramids, one pointing up to the top face center and one pointing down to the bottom face center.
- Each apex sits a height of one half above or below the middle plane.
- A pyramid's volume is one third of base area times height, so each pyramid is one third times one half times one half, which is one twelfth.
- Two of them give one twelfth plus one twelfth, or one sixth.
- So the intersection has volume 1/6, which is answer (D).
💡 Splitting the octahedron at its widest square turns it into two pyramids whose sizes are easy to compute.
5.G.A.2 Put the cube at coordinates from (0,0,0) to (1,1,1). Color the 8 corners like a 8.G.B.7 Take two corners of one tetrahedron, such as (0,0,0) and (1,1,0). They differ by 7.G.A.3 Each face of the cube has two diagonals: one is an edge of Tetra 1 and the other 6.G.A.1 Slice the octahedron by the horizontal plane halfway up the cube, at height one 7.G.B.6 That middle square splits the octahedron into two square pyramids, one pointing Review
Reasonableness: One tetrahedron has volume one third (the unit cube with four corner pieces of volume one sixth each cut off). The overlap must be smaller than a single tetrahedron, and one sixth is indeed less than one third and less than the cube's volume of 1, so the size is sensible. The clean value 1/6 matches choice (D).
Alternative: The six face centers are the midpoints of one tetrahedron's edges, so the octahedron is that tetrahedron's midpoint octahedron. Joining edge midpoints cuts four small corner tetrahedra (each one eighth of the volume) off the big one, leaving half. Half of the tetrahedron's volume one third is one sixth, confirming the answer.
CCSS standards used (min grade 8)
5.G.A.2Represent real-world and mathematical problems by graphing points (Placing the cube on coordinates and labeling the vertices of both tetrahedra.)8.G.B.7Apply the Pythagorean theorem to determine unknown side lengths in right triangles (Showing each tetrahedron edge is a face diagonal of length root two.)7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures (Identifying the shared region as an octahedron with corners at the cube's face centers.)6.G.A.1Find area of triangles, special quadrilaterals, and polygons by composing (Computing the area of the octahedron's middle square from its diagonals.)7.G.B.6Solve real-world problems involving area, surface area, and volume (Adding the two square pyramids to get the intersection volume of one sixth.)
⭐ Two tetrahedra sharing a cube overlap in an octahedron whose corners are the cube's face centers; split it into two square pyramids and the volume is one sixth.
⭐ Two tetrahedra sharing a cube overlap in an octahedron whose corners are the cube's face centers; split it into two square pyramids and the volume is one sixth.
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