AMC 8 · 2025 · #10

Grade 5 geometry-2d
Archetype Overlap / Complement Counting └ Sub-archetype Geometric Area via Inclusion-Exclusion or Area Balance
area-rectanglesspatial-visualizationreflection-symmetry identify-subproblemsarea-difference ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic
📏 Medium solution 💡 3 insights 📊 Diagram
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Problem
A 5 × 3 rectangle ABCD is rotated 90° clockwise about the midpoint M of side DC, producing a second rectangle that overlaps the first. Find the total area (in square inches) that the union of the two rectangles covers.

Pick an answer.

(A)
21
(B)
22.25
(C)
23
(D)
23.75
(E)
25

AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Tool #1 (Draw a Diagram) is the natural first move because the problem is about positions of two overlapping rectangles after a rotation — sketching ABCD, marking the midpoint M, and drawing the rotated copy makes the overlap visible. Tool #10 (Physical Representation) is a great backup if visualization is shaky — cutting a 5 × 3 paper rectangle, marking the midpoint of the long side, and rotating it 90° shows immediately that the overlap is a small square. Tool #7 (Identify Subproblems) then breaks the question into three clean pieces: (i) area of one rectangle, (ii) shape and area of the overlap, (iii) combine via Inclusion-Exclusion: Total = Area₁ + Area₂ - Overlap.

1STEP 1

Sketch rectangle ABCD, then mark M at the middle of DC so that MD = MC = 2.5 in.

MD = MC = 52\frac{5}{2} = 2.5 in
2STEP 2

One rectangle's area is length × width: 5 × 3 = 15 square inches.

Area_one = 5 × 3 = 15 in²
3STEP 3

Fold and rotate a paper copy 90° clockwise about M, and the overlap is a small square of side 2.5 in.

Overlap is a square with side 2.5 in
4STEP 4

The overlap is a 2.5-by-2.5 square, so its area is 2.5 × 2.5 = 6.25 square inches.

Overlap = 2.5 × 2.5 = 6.25 in²
5STEP 5

By inclusion-exclusion, add the two areas and subtract the overlap once: 15 + 15 - 6.25 = 23.75 square inches.

Total = 15 + 15 - 6.25 = 30 - 6.25 = 23.75 in² → (D)
Answer
23.75
Each rectangle alone covers 15 in², so the union must be more than 15 but less than 2 × 15 = 30. Our answer 23.75 sits comfortably between, and the overlap of 6.25 is exactly 12\frac{1}{2} × 2.5 × 5 = 6.25 — half of the 2.5 × 5 "half rectangle," which fits the geometry. Choice (D) is the unique option in that range matching 30 - 6.25.
💡Key takeaway

This AMC 8 problem only needs Grade 5 decimal multiplication you already know — once you see the overlap is a tiny 2.5-by-2.5 square, the rest is just 15 + 15 - 6.25!

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