Problem 6 · 2006 AMC 8
Easy
Geometry & Measurement
perimeter-subtract-overlap
The letter T is formed by placing two 2 × 4 inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?
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Answer: C — 20.
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Hint 1 of 2
When two shapes touch, the edges along the touch line end up INSIDE — they're no longer part of the outline. So find where the two rectangles press together.
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Hint 2 of 2
Total outline = (sum of both perimeters) − (every edge that became internal). The shared seam disappears from BOTH rectangles, so subtract it twice.
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Approach: add the two perimeters, remove the hidden seam
- Each 2 × 4 rectangle has perimeter 2(2 + 4) = 12, so separately they total 24.
- Where the stem meets the bar, a 2-inch segment of each rectangle is pressed against the other. That seam is now interior, so it leaves the outline of both — subtract 2 twice.
- 24 − 2 − 2 = 20.
- Worth keeping: whenever pieces are glued together, perimeter of the whole = sum of the parts' perimeters − 2 × (length of each shared seam). The seam vanishes from both sides, which is why it's counted twice.
Another way — walk the outline directly:
- Trace the T's edge and add the segments: the bar's top is 4; coming down the right side and around the stem and back up the left mixes 2-inch and other pieces.
- Summing all the boundary segments of the T gives 20 — a good way to double-check the subtraction method, since you never touch the interior seam at all.
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