AMC 10 · 2011 · #12

Grade 8 geometry-2dprobability
Archetype Geometric Probability by Area └ Sub-archetype Area-Proportional Region Probability
geometric-probabilityarea-rectanglesarea-triangles identify-subproblems ↑ Prerequisites: area-triangles
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A regular eight-sided board is cut into a centre square, four rectangles, and four corner triangles. Find the chance of hitting the centre.

Pick an answer.

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

Because the dart is uniform, the probability is just one area divided by another, so I never need real measurements — only a ratio. The octagon is an awkward shape, so I break it into pieces I can measure: one center square, four rectangles, four corner triangles. I pick a convenient size for the pieces (a ratio is unaffected by scale), add up the octagon's area, and divide the square's area by it.

1STEP 1

Turn the probability into an area ratio

The probability is a plain area ratio.

P = (area of center square)/(area of octagon)
2STEP 2

Cut the octagon into simple pieces

The cut names nine pieces in all.

octagon = 1 square + 4 rectangles + 4 corner triangles
3STEP 3

Measure a corner triangle and the shared length

One corner triangle fixes the shared length.

hyp = √(1² + 1²) = √(2); 4 × 1/2(1)(1) = 2
4STEP 4

Measure the square and the rectangles

The square and rectangles follow immediately.

square = (√(2))² = 2; 4 rectangles = 4(1 · √(2)) = 4√(2)
5STEP 5

Add up and divide

Dividing and rationalising gives choice (B).

P = 2/(4 + 4√(2)) = 1/(2(1+√(2))) = (√(2)-1)/2
Answer
(√(2) - 1)/2
The center square is one of nine regions and is clearly not the biggest chunk of the board, so a probability well under one third makes sense. Numerically (square root of 2 minus 1) over 2 is about 0.207, roughly one fifth, which matches how small the middle square looks in the figure. The other choices are larger or, like 1/4, do not survive the exact area computation.
💡Key takeaway

For an even target, probability is just the target's area over the whole area — so cut the hard shape into squares, rectangles, and triangles you can measure.

  • Turn the probability into an area ratio
  • Cut the octagon into simple pieces
  • Measure a corner triangle and the shared length
  • Measure the square and the rectangles
  • Add up and divide

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