Party Hats

Geometry Level 3

The proportions of the colored sections relative to the whole circle are 1 2 \dfrac{1}{2} for red, 1 3 \dfrac{1}{3} for pink, and 1 6 \dfrac{1}{6} for yellow, as shown above.

If each section is rolled to make a conic party hat by attaching the edges together (without cutting or overlapping any area), which party hat will be the tallest?

Red Pink Yellow They all have the same height Not enough information

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

Since all hats are made from the same circular sections, they will have the same slant height, which is equivalent to the whole circle's radius R R . Then the formed cone will have the new shorter base radius r r with the new vertical h h .

According to Pythagorean theorem, R 2 = h 2 + r 2 R^2 = h^2 + r^2 .

Since R R is constant for all hats, the smaller r r gets, the higher h h will be.

Then the smaller r r is, the less circumference of the conic base is.

In other words, the cone with least base circumference will be the highest, and from the data given, the yellow section has the least area proportion and, thereby, the least arc length.

As a result, the yellow party hat will be the highest of all.

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