AMC 10 · 2022 · #25
Grade 8 geometry-2d
Pick an answer.
Tool #1 (Diagram) is the unlock, but only in its strongest form: put every corner on coordinates. A picture alone cannot tell you where two slanted hexagon sides cross, and that crossing is exactly what the problem hinges on. Tool #17 (Spatial) supplies the one measurement a regular hexagon of side 1 gives you for free — opposite sides are √(3) apart — which is what makes each hexagon overhang the square. Tool #13 (Algebra) turns the two slanted sides into linear equations and solves for the inward-pointing vertex. Tool #7 (Subproblems) splits the area into a central cross plus four congruent corner pieces, each easy. Tool #16 (Change Focus) is the guard rail: the outline is the union of the four hexagons, not their convex hull, so the four dents must be kept.
Put the square on coordinates
Put the square on coordinates.
Grade 6 coordinate plane — once corners are number pairs, "find the area" becomes arithmetic you can check.
6.G.A.3Draw A DiagramMeasure a hexagon
Measure across its opposite sides.
Grade 8 Pythagoras — half an equilateral triangle of side 1 is a right triangle with legs 1/2 and √(3)/2.
8.G.B.7Visualize Spatial RelationshipsWrite down all four hexagons
Write down all four far sides.
Each hexagon is a √(3)-wide slab laid across a 1-wide square, so it pokes out √(3)-1 on the far side.
6.G.A.3Draw A DiagramLocate the twelve vertices
Locate the twelve outer vertices.
Rotational symmetry means one corner is the whole story — solve it once, spin it three times.
Rotational symmetry means one corner is the whole story: solve it once and spin it around.
▸ Why?
A rotation moves the figure onto itself without stretching, so each corner is an exact copy of the others.
▸ Why?
Four quarter turns fill the whole turn, so the four corners account for the figure with nothing left over.
The two crossing sides
Write the equations of the two crossing sides.
Grade 8 slope — two known points on a hexagon side pin down its entire line.
8.EE.B.6Convert To AlgebraSolve for the inward vertex
Solve for the inward-pointing vertex.
Two lines, one crossing — and its height tells you the boundary caves inward rather than bulging out.
8.EE.C.8Convert To AlgebraArea of the central cross
Compute the central cross's area.
Two overlapping rectangles — add them, then subtract the square you counted twice.
6.G.A.1Identify SubproblemsArea of one corner piece
Compute one corner piece.
The dent splits the corner into two triangles that happen to share the same base length and the same height.
6.G.A.1Identify SubproblemsAdd up and read off
Adding the three numbers gives negative four.
The last trap is reading the question: it wants the three integers added, and p is negative.
8.EE.A.2Change Focus Count The ComplementGrade 8 coordinate geometry handles this AMC 12 closer: a regular hexagon of side 1 is √(3) across, so each hexagon covers the square and pokes out √(3)-1 on the far side. Add the central cross (4√(3)-3) to four corner pieces (3√(3)-5) each and you get 16√(3)-23 — keep the four dents, and remember the question wants m+n+p = -4, not the area.
- Put the square on coordinates
- Measure a side-1 regular hexagon
- Write down all four hexagons
- Locate the 12 outer vertices
- Equations of the two crossing sides
- Solve for the inward vertex
- Area of the central cross
- Area of one corner piece
- Add up, then read off m, n, p
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