AMC 8 · 2008 · #4

Grade 3 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
area-trianglesequal-spacingfraction-arithmetic area-differenceidentify-subproblems ↑ Prerequisites: area-trianglesfraction-arithmetic
📏 Short solution 💡 2 insights 📊 Diagram
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Problem
A big equilateral triangle has area 16. Inside it sits a small equilateral triangle with area 1. The space between the two triangles is split into three congruent trapezoids. What is the area of one trapezoid?

Pick an answer.

(A)
3
(B)
4
(C)
5
(D)
6
(E)
7

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The outer triangle is built from two kinds of pieces: one small triangle plus three trapezoids, with nothing overlapping. Tool #7 (Identify Subproblems) breaks the goal into two easy steps — first find the total area taken by all three trapezoids (subtract the inner triangle from the outer), then split that area evenly among the three congruent trapezoids. Tool #15 (Visualize) confirms the picture: the three trapezoids tile the ring-shaped region between the triangles with no gaps and no overlap, so their areas truly add up to the difference.

1STEP 1

The outer triangle is the inner triangle plus three non-overlapping trapezoids, so its area is the sum of those parts.

outer area = inner area + (area of 3 trapezoids)
2STEP 2

Subtract the inner triangle from the outer triangle to get the three trapezoids' combined area, 15.

area of 3 trapezoids = 16 - 1 = 15
3STEP 3

The three trapezoids are congruent, so split that combined area equally by dividing it by 3.

area of one trapezoid = 153\frac{15}{3} = 5 → (C)
Answer
5
Check the parts add back to the whole: one inner triangle of area 1 plus three trapezoids of area 5 each gives 1 + 3 × 5 = 1 + 15 = 16, which matches the outer triangle. The answer 5 is also in the middle of the choices and a reasonable size — each trapezoid is bigger than the tiny inner triangle but much smaller than the whole figure, just like the picture suggests.
💡Key takeaway

Big triangle minus small triangle gives the trapezoids' combined area — then divide by 3 because they are identical. Subtract, then share equally.

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