AMC 10 · 2013 · #7

Grade 8 geometry-2d
Archetype Similar-Triangle Length / Area Ratio └ Sub-archetype Synthetic Similar-Triangle Length or Area
equal-spacingarea-trianglesthirty-sixty-ninety-triangle systematic-enumerationcasework ↑ Prerequisites: area-trianglesequilateral-triangle
📏 Medium solution 💡 2 insights
Problem
Six points are equally spaced around a circle of radius 1. Choose three of them so the triangle they form is neither equilateral nor isosceles (a scalene triangle, all three sides different). Find the area of that triangle.

Pick an answer.

(A)
$\frac{\sqrt{3}}{3}$
(B)
$\frac{\sqrt{3}}{2}$
(C)
1
(D)
$\sqrt{2}$
(E)
2

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

Drawing the six points as a regular hexagon (tool #1) is what makes the answer visible: it reveals which side is a diameter and lets you read the height off an equilateral triangle. Tool #2 (Make a Systematic List) is used first to pin down the one triangle shape that is scalene by listing the ways the arc gaps can split. Tool #7 (Identify Subproblems) then breaks the area into two clean pieces — find the base, then find the height — instead of chasing all three side lengths at once.

1STEP 1

Find the scalene shape

Three points cut the circle into 60° gap arcs adding to 6; equal gaps give equal sides, so the only all-different split is 6 = 1+2+3.

6 = 2+2+2 = 1+1+4 = 1+2+3
2STEP 2

Spot the diameter

The 3-gap arc is 3 × 60° = 180°, half the circle, so that chord runs through the center: a diameter of length 2, which we take as the base.

3 × 60° = 180°, diameter = 2 × 1 = 2
3STEP 3

Find the height

Joining the third vertex and a base endpoint to the center gives two radii 60° apart — a side-1 equilateral triangle — so h = √(3)/2.

h² + (1/2)² = 1² → h = √(1-1/4) = √(3)/2
4STEP 4

Compute the area

Base 2 and height √(3)/2 give area = 1/2 × 2×√(3)/2 = √(3)/2, which is choice (B).

1/2 × 2×√(3)/2 = √(3)/2 → (B)
Answer
√(3)/2
The triangle fits inside a radius-1 circle, so its area must be well under the circle's area π≈ 3.14; √(3)/2≈ 0.87 is comfortably small. The three sides come out 2, √(3), and 1 — a 30-60-90 right triangle with all sides different, confirming it really is scalene. Choices (D) √(2)≈ 1.41 and (E) 2 are too large for a triangle this size, and (A) √(3)/3≈ 0.58 and (C) 1 do not match the base-height product, so only √(3)/2 survives.
💡Key takeaway

For evenly spaced points on a circle, count the gaps: equal gaps make equal sides, and a chord across half the circle is a diameter you can stand the triangle on.

  • Find the scalene shape
  • Spot the diameter
  • Find the height
  • Compute the area

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