AMC 10 · 2005 · #12

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
circular-sectorequilateral-trianglearea-difference identify-subproblemsarea-difference ↑ Prerequisites: area-circlesequilateral-trianglecircular-sector
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A trefoil is drawn by attaching circular sectors to the sides of congruent equilateral triangles, and it sits on a horizontal base of length 2 that passes straight through its centre. Find the total area of the trefoil.

Pick an answer.

(A)
$\frac{1}{3}\pi+\frac{\sqrt{3}}{2}$
(B)
$\frac{2}{3}\pi$
(C)
$\frac{2}{3}\pi+\frac{\sqrt{3}}{4}$
(D)
$\frac{2}{3}\pi+\frac{\sqrt{3}}{3}$
(E)
$\frac{2}{3}\pi+\frac{\sqrt{3}}{2}$

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The trefoil is a compound shape, so Tool #7 (Identify Subproblems) is the spine: cut it into pieces whose areas are easy. The cleanest cut sits the trefoil on a big equilateral triangle of side 2 (vertices at the two base ends and the top point), then adds four circular "bumps." Each bump is a 60°-sector minus a small equilateral triangle of side 1. Tool #1 (Draw a Diagram) locates those pieces. The payoff comes from Tool #16 (Change Focus): instead of grinding out the messy triangle areas, notice that the four small triangles rebuild exactly the big triangle, so every triangle term cancels and only the sectors survive. Tool #3 (Eliminate Possibilities) then matches the clean result to the choices.

1STEP 1

Fix the radius and the building blocks

The base is a diameter, so every sector radius is 1; a 60° arc plus two radii closes into a side-1 equilateral triangle.

r=1, each arc=60°, each triangle side=1
2STEP 2

Sit the trefoil on a big triangle

Join the base ends to the top: the big equilateral triangle of side 2 is 4 small ones, plus four bumps on its slanted sides.

T_big(side 2)=2² T_small(side 1)=4 T_small
3STEP 3

Measure one bump

Each bump is a segment: the 60° sector of radius 1, area π/6, minus the side-1 triangle under its chord.

sector=1/6π(1)²=π/6, bump=π/6-T_small
4STEP 4

Add the pieces and watch the triangles cancel

Total = 4 small triangles + 4(π/6 − small triangle): the added and subtracted triangles cancel, leaving only sectors.

4T_small+4(π/6-T_small)=4·π/6
5STEP 5

Compute the surviving sectors

Four sixth-of-a-circle sectors remain: 4·π/6=2π/3 — the only choice carrying no square-root term, (B).

4·π/6=2π/3 → (B)
Answer
2/3π
A sanity check: 2/3π≈ 2.09. The big equilateral triangle of side 2 has area √(3)≈ 1.73, and the trefoil is that triangle plus four outward bumps, so its area should be a bit more than 1.73 — and 2.09 is indeed a little larger, which fits. Every other choice adds a √(3) term, but the four added triangles and four subtracted triangles cancel exactly, so a lone π answer is exactly what the geometry predicts; 2/3π is choice (B).
💡Key takeaway

Sit the trefoil on a big triangle and add four pie-slice bumps: the triangles you add and subtract cancel, leaving just 4×1/6 of a circle =2/3π.

  • Fix the radius and the building blocks
  • Sit the trefoil on a big triangle
  • Measure one bump
  • Add the pieces and watch the triangles cancel
  • Compute the surviving sectors

More like this

Same archetype — closest grade level first.