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

In the figure, the outer equilateral triangle has area $16$ , the inner equilateral triangle has area $1$ , and the three trapezoids are congruent. What is the area of one of the trapezoids?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Outer equilateral triangle has area $16$; Inner equilateral triangle has area $1$; The three trapezoids between the triangles are congruent (identical in shape and size); Answer choices: (A) $3$, (B) $4$, (C) $5$, (D) $6$, (E) $7$

Unknowns: The area of one trapezoid

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #15 Visualize

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.

Execute — Answer: C

#15 Visualize 3.MD.C.7 Step 1

Picture the regions.
The outer triangle is made of the inner triangle plus three trapezoids around it, and these pieces do not overlap.
So the area of the outer triangle equals the area of the inner triangle plus the total area of the three trapezoids.

$$\text{outer area} = \text{inner area} + \text{(area of 3 trapezoids)}$$

💡 Grade 3 area work: when a region is split into non-overlapping pieces, the whole area equals the sum of the parts.

#7 Identify Subproblems 3.MD.C.7 Step 2

Find the combined area of all three trapezoids by subtracting the inner triangle's area from the outer triangle's area.

$$\text{area of 3 trapezoids} = 16 - 1 = 15$$

💡 This is the "area of the ring" subproblem: the trapezoids fill exactly the space the small triangle doesn't.

#7 Identify Subproblems 3.OA.A.2 Step 3

The three trapezoids are congruent, so they share the $15$ square units of area equally.
Divide by $3$ to get the area of one trapezoid.

$$\text{area of one trapezoid} = \dfrac{15}{3} = 5 \;\Rightarrow\; \textbf{(C)}$$

💡 Grade 3 partitive division: $15$ shared equally among $3$ identical pieces gives $5$ each.

[1] #15 3.MD.C.7 Picture the regions. The outer triangle is made of the inner triangle plus three

[2] #7 3.MD.C.7 Find the combined area of all three trapezoids by subtracting the inner triangle

[3] #7 3.OA.A.2 The three trapezoids are congruent, so they share the $15$ square units of area

Review

Reasonableness: Check the parts add back to the whole: one inner triangle of area $1$ plus three trapezoids of area $5$ each gives $1 + 3 \times 5 = 1 + 15 = 16$, 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.

Alternative: Tool #5 (Look for a Pattern) using side-length ratios: the outer triangle's area $16$ is $16$ times the inner triangle's area $1$, so the outer side is $\sqrt{16} = 4$ times the inner side. If the inner side is $1$ unit, the outer side is $4$ units and each trapezoid has parallel sides $1$ and $4$. Using the height ratio for equilateral triangles, the trapezoid's area works out to $5$ — same answer (C).

CCSS standards used (min grade 3)

3.MD.C.7 Relate area to the operations of multiplication and addition (Using the fact that the outer triangle's area equals the inner triangle's area plus the combined area of the three trapezoids, so subtraction gives the ring's area.)
3.OA.A.2 Interpret whole-number quotients of whole numbers (Dividing the ring's area of $15$ equally among the $3$ congruent trapezoids to get $15 \div 3 = 5$.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 3)

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