AMC 8 · 2007 · #12

Grade 6 geometry-2d

Archetype Compound Area by Decomposition / Difference

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

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A unit hexagram is composed of a regular hexagon of side length $1$ and its $6$
equilateral triangular extensions, as shown in the diagram. What is the ratio of
the area of the extensions to the area of the original hexagon?

$\mathrm{(A)}\ 1:1 \qquad \mathrm{(B)}\ 6:5 \qquad \mathrm{(C)}\ 3:2 \qquad \mathrm{(D)}\ 2:1 \qquad \mathrm{(E)}\ 3:1$

Pick an answer.

(A)

1:1

(B)

6:5

(C)

3:2

(D)

2:1

(E)

3:1

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: The central shape is a regular hexagon with side length $1$; Six equilateral triangles are attached, one along each side of the hexagon; Each outer triangle shares its base with one full side of the hexagon; Answer choices: (A) $1{:}1$, (B) $6{:}5$, (C) $3{:}2$, (D) $2{:}1$, (E) $3{:}1$

Unknowns: The ratio (area of the $6$ extensions) $:$ (area of the hexagon)

Understand

Plan

Primary tool: #1 Draw a Diagram

Secondary: #5 Spot a Pattern

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 $\sqrt{3}$, no algebra.

Execute — Answer: A

#1 Draw a Diagram 6.G.A.1 Step 1

Slice the hexagon.
Draw the three long diagonals of the regular hexagon — they meet at the center and cut the hexagon into $6$ triangles.
Because the hexagon is regular with side $1$, each of these $6$ triangles is equilateral with side length $1$.

$$\text{Hexagon} = 6 \text{ congruent equilateral triangles, each of side } 1$$

💡 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.

#5 Spot a Pattern 6.G.A.1 Step 2

Identify the extensions.
Each outer extension is equilateral and shares one full side of the hexagon as its base.
That base has length $1$, so every outer extension is an equilateral triangle of side $1$ — the same triangle as each inner wedge.

$$\text{Each extension} = \text{equilateral triangle of side } 1 \;\Rightarrow\; \text{congruent to one inner wedge}$$

💡 Two equilateral triangles with the same side length are always congruent, so they cover equal area.

#5 Spot a Pattern 6.G.A.1 Step 3

Let $T$ stand for the area of one unit equilateral triangle.
The hexagon is $6$ such triangles, and the $6$ extensions are also $6$ such triangles, so both totals are $6T$.

$$\text{Area of hexagon} = 6T,\;\; \text{Area of } 6 \text{ extensions} = 6T$$

💡 Counting equal pieces lets us compare two areas without ever computing $T$ itself.

#5 Spot a Pattern 6.RP.A.1 Step 4

Form the ratio.
The two areas are equal, so the ratio is $1{:}1$.

$$\dfrac{\text{extensions}}{\text{hexagon}} = \dfrac{6T}{6T} = 1 \;\Rightarrow\; 1{:}1 \;\Rightarrow\; \textbf{(A)}$$

💡 Equal counts of congruent pieces means equal area, which means a $1{:}1$ ratio.

[1] #1 6.G.A.1 Slice the hexagon. Draw the three long diagonals of the regular hexagon — they m

[2] #5 6.G.A.1 Identify the extensions. Each outer extension is equilateral and shares one full

[3] #5 6.G.A.1 Let $T$ stand for the area of one unit equilateral triangle. The hexagon is $6$

[4] #5 6.RP.A.1 Form the ratio. The two areas are equal, so the ratio is $1{:}1$.

Review

Reasonableness: 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.

Alternative: Tool #9 (Solve an Easier Related Problem): instead of areas, compare counts. Both the hexagon and the star's outer ring break into the same building block — a unit equilateral triangle. The hexagon uses $6$ of them and the extensions use $6$ of them, so the ratio is $6{:}6 = 1{:}1$. Choice (A).

CCSS standards used (min grade 6)

6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles (Decomposing the regular hexagon into $6$ congruent unit equilateral triangles and recognizing each outer extension as a $7$th copy of the same building block.)
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities (Comparing the two areas as a count of identical pieces and writing the result as the ratio $6{:}6 = 1{:}1$.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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