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Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle) of height $h$ and radius $r$. Here wrap is the natural sense of covering the surface area of the cylinder completely, without cutting the square, creasing however needed. What is the smallest square that suffices for a given $h$ and $r$? For example, a rectangle of dimensions $(h+2r) \times (2 \pi r)$ suffices for how one might wrap a can of tennis balls or a stout candlelike this:

In this $h=3$ and $r=1$ case, the rectangle has dimenions $5 \times 6.28$, and so a square of side $2 \pi$ suffices. But is that optimal?

Merry Christmas!

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Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle) of height $h$ and radius $r$. Here wrap is the natural sense of covering the surface area of the cylinder completely, without cutting the square, creasing however needed. What is the smallest square that suffices for a given $h$ and $r$? For example, a rectangle of dimensions $(h+2r) \times (2 \pi r)$ suffices for how one might wrap a can of tennis balls or a stout candle like this:

In this $h=3$ and $r=1$ case, the rectangle has dimenions $5 \times 6.28$, and so a square of side $2 \pi$ suffices. But is that optimal?

Merry Christmas!

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Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle) of height $h$ and radius $r$. Here wrap is the natural sense of covering the surface area of the cylinder completely, without cutting the square, creasing however needed. What is the smallest rectangle square that suffices for a given $h$ and $r$? For example, a rectangle of dimensions $(h+2r) \times (2 \pi r)$ suffices for how one might wrap a can of tennis balls or a stout candlelike this:

In this $h=3$ and $r=1$ case, the rectangle has dimenions $5 \times 6.28$, and so a square of side $2 \pi$ suffices. But is that optimal?

Merry Christmas!

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