AMC 8 · 2000 · #6

Grade 3 geometry-2d

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

area-rectanglesspatial-visualizationline-symmetry area-differenceidentify-subproblems ↑ Prerequisites: area-rectangles

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A 5 × 5 square ABCD contains three smaller squares with sides 1, 3, 1 arranged so that the 3 × 3 square sits in a corner and the two unit squares sit in adjacent corners. Find the area of the shaded L-shaped region that wraps around the 3 × 3 square.

Pick an answer.

(A)

(B)

(C)

12.5

(D)

(E)

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The labeled lengths 1, 3, 1 pin down every coordinate on the figure, so Tool #1 (Draw a Diagram) lets us read all the side lengths of the L-shape directly off the grid — no algebra needed. Once the L is drawn with its corner coordinates known, Tool #7 (Break Into Subproblems) handles it cleanly: slice the L into two rectangles along a single horizontal or vertical cut, find each rectangle's area, then add. This avoids the larger "big square minus three smaller squares, then divide by 2" route, which is correct but does extra work the diagram itself makes unnecessary.

1STEP 1

Put the figure on a grid: the outer square has side 1 + 3 + 1 = 5, and the shaded L wraps the 3 × 3 square along the left and bottom.

outer side = 1 + 3 + 1 = 5

2STEP 2

One horizontal cut at y = 1 splits the L into two rectangles: a 1 × 3 strip up the left edge and a 4 × 1 strip along the bottom.

top piece: 1 × 3, bottom piece: 4 × 1

3STEP 3

Each area is length × width: the strip is 1 × 3 = 3 and the base is 4 × 1 = 4, so the L's area is 3 + 4 = 7.

L area = 1 × 3 + 4 × 1 = 3 + 4 = 7 → (A)

Answer

Cross-check with the "big minus inner" view. The lower-left 4 × 4 sub-square (corners (0,0) to (4,4)) contains both the L-region and the 3 × 3 middle square exactly, with no leftover. So L = 4 × 4 - 3 × 3 = 16 - 9 = 7. Same answer. A unit-square count also works: drawing the full 5 × 5 grid and counting shaded cells gives 4 along the bottom strip plus 3 along the left strip = 7 — matches (A). Choices (D) 14 and (E) 15 are tempting only if a solver forgets the L wraps around just one 3 × 3 square and instead subtracts all three small squares from 25, getting 25 - 9 - 1 - 1 = 14; that counts the L-region plus its mirror image, not one L alone.

💡Key takeaway

Slice the L into two rectangles along one straight cut. One piece is 1 × 3 = 3, the other is 4 × 1 = 4, and the L's area is 3 + 4 = 7 — answer (A).

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