Cone from its lateral development sector

Text

The lateral surface of a right cone unrolls into a circular sector of radius $15$ cm and central angle $144^\circ$. Find the base radius and the total surface area of the cone.

Solution

The slant height is the sector radius, $a=15$. The sector’s arc length equals the base circle: $2\pi r=\frac{144}{360}\cdot2\pi\cdot15=12\pi\ \Rightarrow\ r=6\ \text{cm}.$ Total surface: $S_t=\pi r(r+a)=\pi\cdot6\cdot21=126\pi\approx395.84$ cm$^2$. $\boxed{r=6\ \text{cm};\ S_t=126\pi\approx395.84\ \text{cm}^2}$