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My book states the following:

Likewise, we can take a circular cone with base radius $r$ and slant height $l$, cut it along the dashed line in Figure 2:

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and flatten it to form a sector of a circle with radius $l$ and central angle $ \theta = \frac{2\pi r}{l} $.

Why does $ \theta = \frac{2\pi r}{l} $?

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The length of the edge of the sector is $2\pi r$ because that is the circumference of the base of the cone.

The radius of the sector is $l$ because that is the distance from the tip of the cone to the base, which becomes the distance from the center of the circle to the edge.

For any arc of length $L$ on a circle of radius $R$, the angle that it subtends is $\frac{L}{R}$ radians, essentially by definition.

Thus, the angle of the sector is $\theta=\frac{2\pi r}{l}$.

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