Competition · AMC preparation · step 4 of 4

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

Slice the hexagon into triangles

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

A regular hexagon's center is the same distance from every vertex as the side length, which is why each of the six wedges is itself equilateral.

Drawing the three long diagonals of a regular hexagon with side length 1 cuts it into six identical equilateral triangles, each with side length 1.

▸ Why?

Cutting from the center out to the six vertices divides the hexagon into six triangles that together fill it with no gaps and no overlaps.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

In each of the six triangles the two sides that run from the center are equal, because both are radii reaching the one circle that passes through all six vertices.

A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.

▸ Why?

The angle at the center inside each triangle is 60 degrees.

▸ Why?

The six center angles are all equal, because turning the hexagon by one sixth of a full turn drops it exactly onto itself and carries each triangle onto the next.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

The six equal center angles together make one full turn of 360 degrees, which is two straight angles placed together, so each angle is 60 degrees.

A straight angle is 180 degreesAngles sitting side by side along a straight line always fill up 180 degrees.

▸ Why?

The other two angles of each triangle are also 60 degrees, so all three of its angles are equal.

▸ Why?

The two base angles are equal, because folding the triangle along the line from the center to the middle of the base lays one base corner exactly onto the other.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

The three angles add up to 180 degrees, so once the 60-degree center angle is taken out the two equal base angles must be 60 degrees each.

A triangle's angles sum to 180 degreesThe three corners of any flat triangle always add up to one straight angle, 180 degrees.

▸ Why?

Since all three angles are equal the three sides are equal too, so each triangle is equilateral, and because its base is one side of the hexagon its every side measures 1.

▸ Why?

In a triangle equal angles are faced by equal sides, so three equal angles force all three sides to be equal.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

The base of each triangle is a full side of the hexagon, which is given as length 1, so the other two equal sides also measure 1.

Transitivity of equalityIf the first equals the second and the second equals the third, the first equals the third.

6.G.A.1Draw A Diagram

2STEP 2

Identify the six extensions

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

Name one triangle's area

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

Form the ratio

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.

Slice the hexagon into triangles
Identify the six extensions
Name one triangle's area
Form the ratio

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Slice the hexagon into triangles

Identify the six extensions

Name one triangle's area

Form the ratio

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