MHB Radian Measure: Show Cone Surface Area is $\pi rl$

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A cone's curved surface area can be derived by transforming it into a sector of a circle with radius equal to the slant height, \( l \), and angle \( \theta \). The relationship between the arc length of the sector and the circumference of the cone's base leads to the equation \( l\theta = 2\pi r \). By calculating the area of the sector, which is given by \( \frac{1}{2}l^2\theta \), and substituting for \( \theta \), it can be shown that the curved surface area of the cone is \( \pi rl \). This derivation highlights the connection between the geometry of the cone and the properties of circular sectors. Understanding this relationship is crucial for solving problems involving cone surface areas.
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A cone with base radius $r$, vertical height $h$ and slant height $l$ has its curved surface slit and flattened out into a sector with radius $l$ and angle $\theta$. By comparing the arc length of this sector with the circumference of the base of the cone, show that $l\theta = 2\pi r$, and deduce by calculating the area of the sector, that the curved surface area of the cone is $\pi rl$.
 
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