AMC 8 · 2007 · #12

Grade 6 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-trianglesratio-proportionsimilar-figures identify-subproblems ↑ Prerequisites: area-triangles

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A unit hexagram is built from a regular hexagon of side length 1 together with 6 equilateral triangles attached to its outside, one on each side. Find the ratio of the total area of the 6 outer triangles to the area of the central hexagon.

Pick an answer.

(A)

1:1

(B)

6:5

(C)

3:2

(D)

2:1

(E)

3:1

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) is the move that cracks this problem: draw the three long diagonals of the hexagon to slice it into 6 congruent equilateral triangles, all of side 1. Once that picture is in front of you, Tool #5 (Spot a Pattern) does the rest — each of the 6 outer extensions is also an equilateral triangle of side 1, so it matches one of the inner pieces exactly. Counting congruent triangles replaces any formula or calculation. No area formula, no √(3), no algebra.

1STEP 1

Draw the three long diagonals: the regular hexagon splits into 6 equilateral triangles, each of side 1.

Hexagon = 6 congruent equilateral triangles, each of side 1

2STEP 2

Each outer extension shares a hexagon side, so it is an equilateral triangle of side 1 — congruent to one inner wedge.

Each extension = equilateral triangle of side 1 → congruent to one inner wedge

3STEP 3

With T the area of one unit triangle, the hexagon is 6T and the 6 extensions are also 6T.

Area of hexagon = 6T, Area of 6 extensions = 6T

4STEP 4

The two areas are equal, so the ratio is 1{:}1 — choice (A).

extensions/hexagon = 6T/6T = 1 → 1{:}1 → (A)

Answer

1:1

Sanity check with a fold: imagine folding each outer point of the star inward along the hexagon side it sits on. Each folded triangle lands exactly on one of the 6 inner wedges with no gap and no overlap — that is only possible if the two triangles are congruent, confirming the area count is equal. The ratio 1{:}1 also matches the symmetry of the figure: there are 6 outer triangles and 6 inner triangles, all the same size. Choices (B) through (E) all claim the extensions cover more area than the hexagon, which the fold argument rules out.

💡Key takeaway

Slice the hexagon into 6 equilateral triangles and the 6 star points are the same triangle — counting congruent pieces gives the area ratio without any formula.

More like this