Question #259994

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 θ\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, α\alpha , as a

function of θ\theta .


1
Expert's answer
2021-11-02T18:04:00-0400

the radius of circular sector = slant height R

surface area of cone = area of a sector


surface area of cone:

A=πr(r+H2+r2)=πr(r+R)A=πr(r+\sqrt{H^2+r^2})=\pi r(r+R)


area of a sector:

(α/360º)πR2=πr(r+R)(\alpha/360º) \cdot πR^2=\pi r(r+R)


for apex angle of the cone:

sin(θ/2)=r/Rsin(\theta/2)=r/R


then:

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


α=sin(θ/2)(sin(θ/2)+1)360°\alpha= sin(\theta/2)(sin(\theta/2)+1)\cdot 360\degree



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