AMC 8 · 2018 · #4

Grade 6 geometry-2d

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

area-trianglesarea-rectanglescoordinate-geometryspatial-visualization area-differenceidentify-subproblems ↑ Prerequisites: area-rectanglesarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A twelve-sided polygon is drawn on 1 cm × 1 cm graph paper with all twelve vertices on grid points. Find the area, in square centimeters, of the region enclosed by the polygon.

Pick an answer.

(A)

(B)

12.5

(C)

(D)

13.5

(E)

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The figure has an awkward 12-sided boundary, but Tool #7 (Identify Subproblems) reveals a clean split: a central 3 × 3 square sitting on the grid from (2,2) to (5,5), plus four congruent right triangles poking outward (one above, one below, one left, one right). Each piece has an area that elementary geometry handles directly — a square as side × side, and a right triangle as $\frac{1}{2}$ × base × height. Tool #1 (Draw a Diagram) is the supporting move: marking the four "point" triangles on the picture and the 3 × 3 square they hug makes the decomposition impossible to miss.

1STEP 1

Plot the 12 vertices and trace the outline: it splits into a central 3 × 3 square from (2,2) to (5,5) with four triangular tips poking out.

Decomposition: 12-gon = (3 × 3 square) + (top tri) + (right tri) + (bottom tri) + (left tri)

2STEP 2

The central square runs from (2,2) to (5,5), so each side is 5 - 2 = 3 cm and its area is 9 cm².

Area_square = 3 × 3 = 9 cm²

3STEP 3

Take the top tip (2,5), (3,6), (4,5): base 4 - 2 = 2 cm and height 1 cm, so its area is 1 cm².

Area_△ =

\frac{1}{2}

× base × height =

\frac{1}{2}

× 2 × 1 = 1 cm²

4STEP 4

By symmetry the right, bottom, and left tips are congruent to the top one, so the four triangles total 4 cm².

4 × 1 = 4 cm²

5STEP 5

Add the central square and the four tips: 9 + 4 gives the enclosed area of 13 cm².

Total area = 9 + 4 = 13 cm² → (C)

Answer

The polygon fits inside a 5 × 5 bounding square from (1,1) to (6,6), so the area is at most 25 and clearly more than the central 3 × 3 = 9. The four small triangular tips look like roughly 1 cm² each by eye, so 9 + 4 = 13 matches the picture. Choice (C) is the only option in that neighborhood.

💡Key takeaway

This AMC 8 problem only needs the Grade 6 triangle-area rule (half of base times height) you already know, plugged into a simple square-plus-four-triangles split!

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