Answer to Question #112089 in Geometry for Alexis Ebuka

Question #112089
A 216 sector of a circle of radius 5cm is bent to form a cone .what is the base radius. What is the vertical angle. What is the curved surface area
1
Expert's answer
2020-04-27T18:23:46-0400

Radius of the circle = R=5cm,R=5cm, angle of the sector = θ=216°.\theta=216\degree.

Length of the circular arc cut off from the circle 

L=2πR×θ360°=2π(5)216°360°=6π(cm)L=2\pi R\times{\theta \over 360\degree}=2\pi(5){216\degree \over 360\degree}=6\pi(cm)

When the sector is cut and its bounding radii is bent to form a cone, slant height of the cone, l=R=5cm.l=R=5cm.

Let rr  and hh be the radius and height of the cone formed.

Circumference of the base of the cone =2πr=6π.=2\pi r=6\pi.


r=3cm.r=3cm.

The base radius is 3cm.3cm.

Height of the cone


h=l2r2=5232=4(cm)h=\sqrt{l^2-r^2}=\sqrt{5^2-3^2}=4(cm)

Let φ\varphi be a vertical angle. Then


sin(φ/2)=rl=35=0.6\sin(\varphi/2)={r \over l}={3\over 5}=0.6

φ=2arcsin(0.6)73.74°\varphi=2\arcsin(0.6)\approx73.74\degree

Find the curved surface area


πrl=π(3)(5)=15π(cm2)47.12(cm2)\pi rl=\pi(3)(5)=15\pi (cm^2)\approx47.12(cm^2)


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