Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #6
Grade 3 geometry-2d
Pick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Label the T's sides
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.
Drawing the shape and writing each length next to its edge turns the problem into simple addition along the outline.
3.MD.D.8Draw A DiagramTrace the boundary edges
Walk clockwise around the outline, listing each edge in order — the lengths come out as 4, 2, 1, 4, 2, 4, 1, 2 inches.
Going in one direction around the shape guarantees each outside edge is counted exactly once.
Tracing the T's outline once from a corner in a single direction meets each of its eight outer edges exactly once, so adding those eight edge lengths gives the perimeter.
▸ Why?
The perimeter is just the total length of the figure's outer boundary, so if you list every piece of that boundary once, adding the pieces gives the perimeter.
▸ Why?
The outer boundary is one closed loop cut into straight edges that touch only at their corners, so the edges fit together with no gap or overlap and add back to the whole boundary.
▸ Why?
Going around the loop in one direction and stopping when you return to the start matches each edge with exactly one leg of the walk, so no edge is skipped and none is counted twice.
Add the eight edges
Add the eight edge lengths: 4+2+1+4+2+4+1+2 = 20 inches → choice (C).
Grouping the edges makes the arithmetic clean: top + bottom = 4 + 2 = 6; the two long stem sides = 4 + 4 = 8; the two short sides of the horizontal bar = 2 + 2 = 4; the two overhangs = 1 + 1 = 2. Total = 6 + 8 + 4 + 2 = 20.
3.MD.D.8Draw A DiagramFor 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.
- Label the T's sides
- Trace the boundary edges
- Add the eight edges
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