Problem

If there is a cone, its generating line is $a$, and the slope angle is $α$, find the area of the side surface.

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Solution

The side surface of this cone is fan-shaped with radius $AC$, and if its central angle is $θ$, its area $S$ is

$$S=\frac{πa^2×θ}{360°}.$$

The length of this fan-shaped arc is equal to the circumference of the base circle, but the length of the radius $AD$ of that circle is

$$\frac{AD}{AC}=\frac{AD}{a}=cos⁡α,$$

$$∴ \ AD=a×cos⁡α.$$

Therefore, the circumference of the bottom is

$$2AD×π=2πa×cos⁡α,$$

$$∴ \ \frac{θ}{360°}=\frac{2πa×cos⁡α}{2πa}=cos⁡α.$$

Thus, the area of this sector (side surface of the cone) is

$$S=πa^2×cos⁡α.$$

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Reference

Yoshikazu Yamakawa, ed. (1997) Okayama ken no Sangaku (Sangaku in Okayama Prefecture), pp.48-49; p.314.