19. Surface area of a circular cone

수학 모음 (Maths collection) 전체보기

 

To find the surface area of a circular cone, we need to consider the area of the circular sector and the area of the circle.

 

 

• The area of the circular sector: We find the angle $\theta$ by considering the arc length, i.e. $$\begin{align} L=2\pi\ell\times\frac{\theta}{360}&=2\pi r \\ \Rightarrow\quad \ell\times\frac{\theta}{360}&=r \end{align}$$ The area is then $$\begin{align} A_{\rm sector}&=\pi\ell^2\times\frac{\theta}{360} \\ &=\pi\ell\times\underbrace{\left(\ell\times\frac{\theta}{360}\right)}_{r} \\ &=\pi r\ell \end{align}$$
• The area of the circle: $$\begin{align} A_{\rm circle}=\pi r^2 \end{align}$$

 

This gives: $$\begin{align} A&=A_{\rm circle}+A_{\rm sector} \\ &=\pi r^2+\pi r\ell \\ &=\pi r(r+\ell) \qquad\square \end{align}$$