AMC 8 · 2006 · #6

Grade 3 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border

perimeterarea-rectanglesspatial-visualization area-differenceidentify-subproblems ↑ Prerequisites: perimetermulti-digit-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

Two 2 × 4 inch rectangles are placed against each other to form the letter T — one rectangle laid flat across the top, the other standing upright underneath, centered. What is the perimeter of the resulting T shape?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Picture

Tool #1 (Draw a Picture) is the natural first move for any perimeter question on an irregular shape: sketch the T, label every side length, then walk around the boundary once and add the segments. The T has eight straight edges, each of which is either 1, 2, or 4 inches, all readable straight from the figure. Tool #7 (Break into Subproblems) gives a clean check: treat the T as the two rectangles, add their perimeters, then subtract the part of the boundary that gets hidden where they meet.

1STEP 1

Sketch and label the T's edges. The upright piece (2 wide) is centered under the 4-wide top, so the bar overhangs 1 inch on each side.

overhang on each side =

\frac{4 - 2}{2}

= 1 inch

2STEP 2

Walk clockwise around the outline, listing each edge in order — the lengths come out as 4, 2, 1, 4, 2, 4, 1, 2 inches.

edges in order = 4, 2, 1, 4, 2, 4, 1, 2 inches

3STEP 3

Add the eight edge lengths: 4+2+1+4+2+4+1+2 = 20 inches → choice (C).

P = 4 + 2 + 1 + 4 + 2 + 4 + 1 + 2 = 20 inches → (C)

Answer

Two separate 2 × 4 rectangles would each have perimeter 2(2+4) = 12, for a combined 24 inches. When the rectangles are pushed together to form the T, a 2-inch segment of each rectangle's edge is hidden inside the figure, so the perimeter loses 2 + 2 = 4 inches. That gives 24 - 4 = 20 inches, matching the boundary-trace answer. The result 20 is one of the offered choices and sits between 16 and 24, exactly where it should be for a shape made of two joined rectangles.

💡Key takeaway

For a perimeter question on an irregular shape, draw the figure, label every edge, and walk around the outside once — adding 4+2+1+4+2+4+1+2 gives 20.

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