AMC 8 · 2012 · #23

Grade 7 geometry-2d

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

perimeterarea-trianglessimilar-figuresratio-proportion identify-subproblemseasier-related-problem ↑ Prerequisites: perimeterarea-trianglesratio-proportion

📏 Medium solution 💡 3 insights

Problem

An equilateral triangle and a regular hexagon have the same perimeter. The triangle has area 4. Find the area of the hexagon.

Pick an answer.

(A)

(B)

(C)

(D)

$4\sqrt{3}$

(E)

$6\sqrt{3}$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Diagram) is the natural first move for a 2D geometry problem with no figure given — drawing the hexagon reveals a key fact: a regular hexagon is exactly 6 small equilateral triangles joined at the center. That turns the hexagon's area into a sum we can control. Tool #7 (Subproblems) splits the work into two clean pieces: (a) compare the side of one small triangle to the side of the big triangle, and (b) use the side ratio to get the area of one small triangle, then multiply by 6. Tool #9 (Easier Problem) covers the side-to-area scaling: equilateral triangles are all similar, so doubling the side multiplies the area by 2² = 4 — no square-root-of-3 formula needed.

1STEP 1

Equal perimeters give 3 s_t = 6 s_h, so s_t = 2 s_h — the triangle's side is twice the hexagon's.

3 s_t = 6 s_h → s_t = 2 s_h

2STEP 2

Cut the hexagon from its center into 6 congruent equilateral triangles of side s_h, so its area is six small triangles.

A_hex = 6 · A_small

3STEP 3

Both triangles are similar with side ratio 2, so areas scale by 2² — the big triangle is 4 times a small one.

\frac{A_big}{A_small}

= (

\frac{s_t}{s_h}

)² = 2² = 4

4STEP 4

The big triangle's area 4 is 4× a small one, so each small triangle has area 1.

A_small =

\frac{A_big}{4}

\frac{4}{4}

= 1

5STEP 5

Six small triangles of area 1 fill the hexagon, so its area is 6 — choice (C).

A_hex = 6 · 1 = 6 → (C)

Answer

The big triangle covers area 4 with 3 long sides; the hexagon has 6 shorter sides (s_h =

\frac{1}{2}

s_t) but more of them, and it is much closer to a circle than a triangle is. With the same perimeter, the more circle-like shape should enclose more area — so the hexagon's area being larger than 4 matches intuition. Answer 6 also matches the "hexagon = 6 unit triangles" picture exactly. The other choices fail this picture: 4 (no gain at all), 5 (does not split into 6 equal pieces of area 1), and the √(3) choices come from misusing the

\frac{√(3)}{4}

s² formula and forgetting that we never need a numeric side length here.

💡Key takeaway

Cut the hexagon into 6 small equilateral triangles — the big triangle has the same shape but doubled side, so its area is 4 times one small triangle. That makes each small triangle area 1, and 6 × 1 = 6.

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