Answer to Question #259994 in Geometry for Sayem

Question

Imagine a right circular cone with altitude H and slant height R. Let the radius of

the circular base be r. Imagine that the apex angle of the cone is $\theta$ . Now slit the

cone open by cutting along the slant height from the base to the apex. Unwrap the

slit cone to reveal a circular sector. Find the central angle of this sector, $\alpha$ , as a

function of $\theta$ .

Answer 1

the radius of circular sector = slant height R

surface area of cone = area of a sector

surface area of cone:

$A=πr(r+\sqrt{H^2+r^2})=\pi r(r+R)$

area of a sector:

$(\alpha/360º) \cdot πR^2=\pi r(r+R)$

for apex angle of the cone:

$sin(\theta/2)=r/R$

then:

$(\alpha/360º) = sin(\theta/2)(sin(\theta/2)+1)$

$\alpha= sin(\theta/2)(sin(\theta/2)+1)\cdot 360\degree$