a 216 degrees sector of radius 5cm is bent to form a cone. find the radius of the cone
Answers
Answer:
Radius of the circle = R=5cm,R=5cm, angle of the sector = \theta=216\degree.θ=216°.
Length of the circular arc cut off from the circle
L=2\pi R\times{\theta \over 360\degree}=2\pi(5){216\degree \over 360\degree}=6\pi(cm)L=2πR×
360°
θ
=2π(5)
360°
216°
=6π(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 =2\pi r=6\pi.=2πr=6π.
r=3cm.r=3cm.
The base radius is 3cm.3cm.
Height of the cone
h=\sqrt{l^2-r^2}=\sqrt{5^2-3^2}=4(cm)h=
l
2
−r
2
=
5
2
−3
2
=4(cm)
Let \varphiφ be a vertical angle. Then
\sin(\varphi/2)={r \over l}={3\over 5}=0.6sin(φ/2)=
l
r
=
5
3
=0.6
\varphi=2\arcsin(0.6)\approx73.74\degreeφ=2arcsin(0.6)≈73.74°
Find the curved surface area
\pi rl=\pi(3)(5)=15\pi (cm^2)\approx47.12(cm^2)πrl=π(3)(5)=15π(cm
2
)≈47.12(cm
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The base radius of the cone is 3cm.
Given:
Ø = 216°
r = 5cm
When a circle is bent to form a cone, the Length of Arc become the base circle of the cone while
r = L = 5cm
To Find:
Radius of the cone
Solution:
Length of arc = ∅/360 x 2πr
Length of the arc of the circle = 216/360 x 2πx5 =18.85 cm
The length of arc = circumference of the base cone
18.85 = 2πr
18.85/2π = r
r= 3cm
Therefore, the base radius of the cone is 3cm.
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