AMC 8 · 2025 · #12

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length

area-circlescoordinate-geometrypythagorean-theoremline-symmetry coordinate-geometryidentify-subproblems ↑ Prerequisites: area-circlescoordinate-geometrypythagorean-theorem

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A plus-shaped region is made of 24 unit squares (each side 1 cm), symmetric horizontally and vertically. We want the area, in square centimeters, of the largest circle that fits entirely inside this region (it may touch the boundary).

Pick an answer.

(A)

$3\pi$

(B)

$4\pi$

(C)

$5\pi$

(D)

$6\pi$

(E)

$8\pi$

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

How to solve

Strategy Draw a Diagram

The figure is geometric and symmetric, so Tool #1 (Draw a Diagram) — overlaying a coordinate grid on the plus shape — instantly reveals the symmetry center and the boundary vertices to watch. Tool #7 (Identify Subproblems) splits the question into three clean pieces: (a) Where is the circle's center? (b) What is the radius? (c) What is the area? Each piece is easy by itself. Tool #3 (Eliminate Possibilities) is the safety net: once we find r² = 5, the answer must be 5π, knocking out the other four choices immediately.

1STEP 1

By symmetry the largest inscribed circle must share the figure's center, at C = (3, 4).

C = (3, 4)

2STEP 2

The concave inner corners nearest C, such as (5, 5), pinch the circle more than the outer edges — so one of them sets the radius.

Inner corners near C: (5,5), (4,6), (2,6), (1,5), …

3STEP 3

Each nearest corner is 2 across and 1 up from C, so by the Pythagorean theorem r² = 2² + 1² = 5 and r = √(5) cm.

r = √((5-3)² + (5-4)²) = √(4 + 1) = √(5)

4STEP 4

The distance to the outer edges is 3, and √(5) ≈ 2.24 ＜ 3, so they don't bind — the circle is tangent to all 8 inner corners at once.

√(5) ≈ 2.236 ＜ 3 (distance to outer edge)

5STEP 5

Apply A = π r²: squaring √(5) gives 5, so the area is exactly 5π square cm — choice (C).

A = π r² = π (√(5))² = 5π → (C)

Answer

5π

Sanity check: the plus shape contains 24 unit squares, so its total area is 24 square centimeters. Our circle's area is 5π ≈ 15.7 square cm — comfortably less than 24 and clearly bigger than a circle inscribed in just the central 2× 2 block (which would have area π ≈ 3.14). The radius √(5) ≈ 2.24 cm also makes sense: the central part of the region extends 3 units out from C horizontally and vertically, so a circle of radius ≈ 2.24 comfortably fits while being pinched by the concave corners — exactly what we found.

💡Key takeaway

This AMC 8 problem only needs the Grade 8 Pythagorean-theorem distance formula (plus the Grade 7 circle-area formula A = π r²) you already know!

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