AMC 10 · 2010 · #2

Grade 4 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border
area-rectanglesarea-differencelinear-equations-one-var identify-subproblemsarea-difference ↑ Prerequisites: area-rectangles
📏 Medium solution 💡 2 insights 📊 Diagram
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Problem
A six-sided figure with every corner square has only four of its sides labelled. Find the area it encloses.

Pick an answer.

(A)
22
(B)
24
(C)
26
(D)
28
(E)
30
How to solve
Strategy Identify Subproblems

An L has no area formula of its own, but it is built out of rectangles, which do — so Tool #7 (Identify Subproblems) says cut it into rectangles and add. The obstacle comes first. Two of the six sides carry no number, and every cut you could make needs one of them, so nothing can be multiplied until those two lengths are pinned down. Tool #4 (Introduce a Variable) names them and Tool #13 (Convert to Algebra) forces their values from the one structural fact available: the boundary closes. Once the shape is fully measured, Tool #16 (Change Focus / Count the Complement) recomputes the area from the outside in — fill the L out to a full rectangle and subtract the bite that is missing — as a check on the cut-and-add count.

1STEP 1

Name the two unlabeled sides

Two sides are unlabelled, so name them.

a=inner horizontal side, b=inner vertical side
2STEP 2

Close the loop to find a and b

Closing the loop recovers both lengths.

5 = a + 2 → a = 3, and 2 + b = 8 → b = 6
3STEP 3

Cut the L into two rectangles

Cutting into two rectangles gives 22.

2 × 8 + 3 × 2 = 16 + 6 = 22
4STEP 4

Check by filling out the rectangle

Filling out the big rectangle confirms 22.

5 × 8 - 3 × 6 = 40 - 18 = 22 → (A)
Answer
22
Bounds alone are not enough here: the L sits inside the 5 × 8 rectangle and contains the 2 × 8 tall arm, so its area is between 16 and 40 — and every answer choice satisfies that. So count unit squares instead of multiplying. The two bottom rows, from height 0 to 1 and from 1 to 2, are 5 squares wide, giving 10 squares. The rows above them, from height 2 up to 8, are only 2 squares wide, and there are 6 of them, giving 12. The tally is 10 + 12 = 22 squares, reached by counting rather than by any area formula. It matches.
💡Key takeaway

When a figure hides some of its side lengths, walk all the way around it — the trip has to close, and that alone hands you the missing lengths.

  • Name the two unlabeled sides
  • Close the loop to find a and b
  • Cut the L into two rectangles
  • Check by filling out the rectangle

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