AMC 10 · 2013 · #15

Grade 8 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
equilateral-trianglearea-regular-hexagonratio-proportion convert-to-algebraidentify-subproblems ↑ Prerequisites: area-trianglesperimeter
📏 Medium solution 💡 2 insights
Problem
A wire is cut into two pieces of length a and b. The piece of length a is bent into an equilateral triangle, and the piece of length b is bent into a regular hexagon. The two shapes have equal area. Find the ratio a/b.

Pick an answer.

(A)
1
(B)
$\frac{\sqrt{6}}{2}$
(C)
$\sqrt{3}$
(D)
2
(E)
$\frac{3\sqrt{2}}{2}$

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

The words "equal area" are really an equation in disguise, so the main move (tool #13) is to write each area in terms of a and b and set them equal. Tool #7 (Identify Subproblems) splits the work into two clean pieces — find the triangle's area, then the hexagon's area — instead of juggling both shapes at once. Tool #4 (Introduce a Variable) turns each perimeter into a side length (a/3 and b/6) so the areas become expressions we can compare.

1STEP 1

Turn wire lengths into sides

Wire a is the triangle's perimeter, so each side is a/3; wire b is the hexagon's perimeter, so each side is b/6.

triangle side = a/3, hexagon side = b/6
2STEP 2

Area of the triangle

A side-s equilateral triangle has area √(3)/4s², so putting in s=a/3 gives the triangle's area √(3)a²/36.

A_△=√(3)/4(a/3)²=√(3) a²/36
3STEP 3

Area of the hexagon

The hexagon is 6 small equilateral triangles of side b/6, so its area is 6·√(3)/4(b/6)², which simplifies to √(3)b²/24.

A_hex=6·√(3)/4(b/6)²=6√(3) b²/(4 · 36)=√(3) b²/24
4STEP 4

Set the areas equal

Equal area means √(3)a²/36=√(3)b²/24; cancel √(3) and cross-multiply to get a²/b²=3/2.

√(3) a²/36=√(3) b²/24 → a²/36=b²/24 → a²/b²=36/24=3/2
5STEP 5

Take the square root

Take the positive square root: a/b=√(3/2)=√(6/4)=√(6)/2, which is choice (B).

a/b=√(3/2)=√(6/4)=√(6)/2 → (B)
Answer
√(6)/2
The answer √(6)/2≈ 1.22 is a bit more than 1, which makes sense: for equal areas the triangle (a less efficient shape) needs a little more wire than the hexagon, so a ＞ b and a/b ＞ 1. Checking size: (a/b)²=6/4=3/2 exactly matches the area equation, and no other choice squares to 3/2 — (A) gives 1, (C) gives 3, (D) gives 4, (E) gives 9/2 — so only (B) fits.
💡Key takeaway

Both shapes are built from the same little equilateral triangle, so "equal area" becomes one clean equation — set the areas equal, cancel, and square-root to get the ratio.

  • Turn wire lengths into sides
  • Area of the triangle
  • Area of the hexagon
  • Set the areas equal
  • Take the square root

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