AMC 8 · 2007 · #23

Grade 6 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-trianglesarea-rectanglesspatial-visualization area-differenceidentify-subproblems ↑ Prerequisites: area-trianglesarea-rectangles

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A 5 × 5 unit grid contains a shaded pinwheel figure. The pinwheel's four blades meet at the center of the grid, and the figure's outline is drawn on grid lines and lines through the center. Find the area of the shaded pinwheel, in square units.

Pick an answer.

(A)

: 4

(B)

: 6

(C)

: 8

(D)

: 10

(E)

: 12

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

The pinwheel has slanted edges and a complicated outline, but the unshaded region is much friendlier — it's just four corner unit squares and four side triangles. That is the signature of Tool #16 (Count the Complement): measure what is easy and subtract from the whole. Tool #1 (Draw a Diagram) sets the bottom-left corner at (0,0) so the center sits at (2.5, 2.5) and every blade tip has clean coordinates. Tool #7 (Identify Subproblems) breaks the unshaded region into two simple pieces — four 1 × 1 squares and four congruent triangles — each of which falls to a basic Grade 6 area formula.

1STEP 1

Place the grid's bottom-left corner at (0, 0), so the center is (2.5, 2.5) and the total area is 5 × 5 = 25.

Total area = 5 × 5 = 25

2STEP 2

At each corner the pinwheel leaves an uncovered 1 × 1 square, so the four corner squares total 4.

4 × (1 × 1) = 4

3STEP 3

The bottom triangle has base 3 (from (1,0) to (4,0)) and height 2.5 (up to the center), so its area is 3.75.

Area of one triangle =

\frac{1}{2}

× 3 × 2.5 = 3.75

4STEP 4

By the pinwheel's four-fold symmetry the other three side triangles match the bottom one, so the four together total 15.

4 × 3.75 = 15

5STEP 5

Subtract the unshaded total 4 + 15 = 19 from the grid's 25: 25 - 19 = 6, the shaded pinwheel's area.

Shaded area = 25 - (4 + 15) = 25 - 19 = 6 → (B)

Answer

: 6

The shaded pinwheel clearly fits inside the 5 × 5 grid, so its area must be less than 25. Picture each blade as roughly the size of a 1 × 1.5 strip narrowed near the center — four blades that small total a few square units, so something around 6 feels right. Answer 6 also gives a clean split: shaded 6, unshaded 19, sum 25. The other answer choices give awkward complements (4 → 21, 8 → 17, 10 → 15, 12 → 13); none break into a simple "four squares + four congruent triangles" picture the way 6 does.

💡Key takeaway

The pinwheel itself has slanted edges, but the empty space around it is plain: four unit squares and four equal triangles. Add those up, subtract from 25, and the pinwheel's area falls out as 6.

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