AMC 10 · 2003 · #15

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation
isosceles-right-trianglearea-differenceratio-proportion identify-subproblems ↑ Prerequisites: area-triangles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A regular octagon ABCDEFGH has area 1. The four vertices A, B, E, F form a rectangle. Find the area of that rectangle.

Pick an answer.

(A)
$1-\frac{\sqrt2}{2}$
(B)
$\frac{\sqrt2}{4}$
(C)
$\sqrt2-1$
(D)
$\frac{1}2$
(E)
$\frac{1+\sqrt2}{4}$
How to solve
Strategy Identify Subproblems

Comparing a rectangle to a whole octagon looks hard until you cut both into pieces you already know. Tool #1 (Draw a Diagram) reframes the octagon as a big square with its four corners sliced off — corners that are 45-45-90 triangles. Tool #4 (Introduce a Variable) names the octagon's side length s so every length becomes a formula. Then tool #7 (Identify Subproblems) splits the job into three easy area calculations: the octagon (square minus four corners), the rectangle (width times height), and their ratio. Since only the ratio matters, the messy √2 terms cancel and the answer falls out cleanly.

1STEP 1

See the octagon as a cut-corner square

The octagon is a square with four equal corner triangles cut off at 45 degrees.

octagon = (big square) - (four equal corner triangles)
2STEP 2

Name the side and find the corner legs

Each corner is a 45-45-90 triangle whose hypotenuse is the octagon side, so its legs are s over root two.

leg = s/√2
3STEP 3

Compute the octagon's area

Subtracting the corners from the square gives the octagon as 2s²(1+√2).

octagon = s²(3+2√2) - s² = s²(2+2√2) = 2s²(1+√2)
4STEP 4

Compute the rectangle's area

The rectangle is one side across by the octagon's full height, or s²(1+√2).

rectangle = s · s(1+√2) = s²(1+√2)
5STEP 5

Compare the two areas

Dividing cancels everything, so the rectangle is half the octagon: 1/2, choice (D).

octagon/rectangle = (2s²(1+√2))/(s²(1+√2)) = 2 → rectangle = 1/2 · 1 = 1/2 → (D)
Answer
1/2
Half is a believable answer: the rectangle ABEF is the tall middle strip of the octagon, and the two leftover pieces on its left and right sides are mirror images of each other, so together they should roughly balance the middle strip — a split near half. A quick numerical check confirms it: with the octagon drawn on a unit circle its area is 2√2 ≈ 2.83 and the rectangle is √2 ≈ 1.41, whose ratio is exactly 2. So the rectangle is half of 1, namely 1/2.
💡Key takeaway

Turn the octagon into a square with trimmed corners, write every length in terms of one side s, and the √2 pieces cancel to show the rectangle is exactly half.

  • See the octagon as a cut-corner square
  • Name the side and find the corner legs
  • Compute the octagon's area
  • Compute the rectangle's area
  • Compare the two areas

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