AMC 10 · 2005 · #20

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
area-differencearea-trianglespythagorean-theorem identify-subproblemsarea-difference ↑ Prerequisites: area-rectanglesarea-trianglesangle-sum-triangle
📏 Medium solution 💡 3 insights
Problem
An eight-sided figure has all of its angles equal. Four of its sides have length 1 and the other four have length √(2)/2, and the two lengths take turns so that no side sits next to another of its own length. Find the area inside the figure.

Pick an answer.

(A)
$\frac{7}{2}$
(B)
$\frac{7\sqrt{2}}{2}$
(C)
$\frac{5+4\sqrt{2}}{2}$
(D)
$\frac{4+5\sqrt{2}}{2}$
(E)
7

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

A shape problem cries out for a picture, so Tool #1 (Draw a Diagram) leads. Once the octagon is drawn with every angle marked 135°, the four short sides tilt at exactly 45°. That tilt is the key: if I extend the four long sides until they meet, they trap the octagon inside a square, and the only thing sticking out beyond the octagon are four little corner triangles. So instead of measuring the awkward eight-sided region directly, I use Tool #16 (Change Focus / Count the Complement): area of octagon = area of the square - the four corners. Tool #7 (Identify Subproblems) then splits the job into two clean pieces I already know how to do — the area of a square and the area of a right triangle.

1STEP 1

Every angle is 135 degrees

An octagon's angles add to 1080°, and all eight are equal, so each is 135° — every corner turns by 45°, tilting the short sides.

((8-2) · 180°)/8 = 1080°/8 = 135°
2STEP 2

Box the octagon in a square

Extend the four long sides until they meet: they close into a square whose only excess is four small corner triangles.

[octagon] = [square] - 4 [corner triangle]
3STEP 3

Each corner triangle is a 45-45-90 with legs 1/2

Each short side is a corner triangle's hypotenuse, so 2ℓ²=(√(2)/2)²=1/2 and every leg is ℓ=1/2.

2ℓ²=(√(2)/2)²=1/2 → ℓ²=1/4 → ℓ=1/2
4STEP 4

Find the square's area and one corner's area

The square's side is leg + long side + leg = 1/2+1+1/2=2, so its area is 4, while each corner triangle covers only 1/8.

[square]=2²=4, [corner triangle]=1/2·1/2·1/2=1/8
5STEP 5

Subtract the four corners

The four corners total 4·1/8=1/2, so the octagon is 4-1/2=7/2 — choice (A).

[octagon]=4-4·1/8=4-1/2=7/2 → (A)
Answer
7/2
The bounding square has area 4, and the octagon is the square with four small corners shaved off, so its area must be a little less than 4 — and 7/2=3.5 is exactly that, which passes the sanity check. A second check: place the octagon on a grid with a bottom-left vertex at (0,0) and walk the sides, landing on vertices (1,0),(1.5,0.5),(1.5,1.5),(1,2),(0,2),(-0.5,1.5),(-0.5,0.5); the shoelace formula on these points gives 1/2 · 7 = 7/2, confirming the answer independently.
💡Key takeaway

Trap the octagon inside a square, then subtract the four little corner triangles: 4-1/2=7/2.

  • Every angle is 135 degrees
  • Box the octagon in a square
  • Each corner triangle is a 45-45-90 with legs 1/2
  • Find the square's area and one corner's area
  • Subtract the four corners

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