AMC 10 · 2014 · #13

Grade 8 geometry-2d

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

equilateral-trianglearea-trianglespythagorean-theorem identify-subproblemsspatial-visualization ↑ Prerequisites: area-trianglesequilateral-triangle

📏 Long solution 💡 3 insights 📊 Diagram

Problem

Equilateral triangle

ABC

has side length 1, and squares

ABDE

BCHI

CAFG

are built outward on its three sides. Their six outer corners form a hexagon. Find the area of hexagon

DEFGHI

Pick an answer.

(A)

$\dfrac{12+3\sqrt3}4$

(B)

$\dfrac92$

(C)

$3+\sqrt3$

(D)

$\dfrac{6+3\sqrt3}2$

(E)

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The hexagon is a compound shape. Instead of chasing its outer coordinates, cut it into pieces whose areas are easy: the central triangle, the three squares, and one small triangle tucked in at each vertex. Add the pieces back up.

1STEP 1

Cut the hexagon into pieces

The hexagon splits cleanly into the middle triangle $ABC$ , the three squares, and three corner triangles filling the gaps.

[DEFGHI] = [ABC] + 3 [square] + 3 [corner triangle]

2STEP 2

Area of the central triangle

An altitude splits triangle $ABC$ into 30-60-90 halves, so its area is $\frac{\sqrt{3}}{4}$ .

h=√(1²-(1/2)²)=√3/2, [ABC]=1/2 · 1·√3/2=√3/4

3STEP 3

Area of the three squares

Each square stands on a side of length 1, so each has area 1, and the three together total $3$ .

3×(1 · 1)=3

4STEP 4

Find the corner angle

The four angles at vertex $A$ fill $360^\circ$ , so the corner angle left over is $120^\circ$ .

∠ EAF = 360°-60°-90°-90° = 120°

5STEP 5

Area of one corner triangle

Splitting the $120^\circ$ wedge $AEF$ (legs 1) gives area $\frac{\sqrt{3}}{4}$ each, so three come to $\frac{3\sqrt{3}}{4}$ .

[AEF]=1/2·√3_base·1/2_height=√3/4, 3 [AEF]=3√3/4

6STEP 6

Add the pieces

Adding the pieces, the two root-three parts merge into one, so the total is $3+\sqrt{3}$ .

√3/4+3+3√3/4=3+√3

Answer

3+√3

Rough size check: three unit squares alone give 3, and the leftover triangles are all thin slivers, so the total should sit a little above 3. The answer 3 + root three is about 4.73, which fits. Choices like 6 or 9/2 would need far more filler area than the small corner triangles provide.

💡Key takeaway

Slice a strange shape into a triangle, some squares, and the little triangles filling the corner gaps, then add every piece.

Cut the hexagon into pieces
Area of the central triangle
Area of the three squares
Find the corner angle
Area of one corner triangle
Add the pieces