Question

A sector of central angle 216 degree is cut from a circle of radius 25 cm and is rolled up into a cone. What are the base radius and height of the cone ? What is its volume

Answer 1

Step-by-step explanation:

Radius of the circle = R = 25 cm.

angle of the sector = Ф = 216°.

Length of the circular arc cut off from the circle 

  = 2 π R * Ф/360°

  = 2 * 22/7 * 25 * 216/360  cm

 =  94.29 cm

The piece is bent into a right circular cone.

So the slanting height s of the cone = radius of circle = R = 25 cm.

s = 25 cm

circumference of the base of the cone = arc length = 94.29 cm

base radius r of the cone = r = 94.29 * 1/2π = 148.17 cm

height of the cone = h = √(s² - r²) = 25² - 148.17²) =  -21329.35 cm 

Volume of the cone = V = π/3 * r² h 

 V = 22/7 * 1/3 * 148.17² * -21329.35 cm³

   = −1.046355326×10¹³ cm³

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Answer 2

The radius of the Cone is 15 cm, the height is 20 cm and the volume is 1500π cm³.

Given:

A sector of central angle 216° is cut from a circle of radius 25 cm and is rolled up into a cone.

To find:

What are the base radius and height of the cone? What is its volume?

Solution:

Formula used:

The arc length of the sector is: (Arc Measure / 360°) ⋅ 2πr

From the data,

A sector of central angle 216° is cut from a circle  

Length of the sector = [ 216/360 ] × 2π(25) = 30π

Here the sector is formed as a cone

Length of the Arc = Circumference of the base of the cone

Let R be the radius of the cone  

=> 30π = 2πR

=> R = 15 cm  

∴ The radius of the Cone is 15 cm

 

Here slant height of the cone will equal the radius of the original circle

Height of the cone can be calculated as follows

Height = √ [ Slant height² - Radius² ]

Height = √25² - 15² = 20 cm

∴ The height of the cone is 20 cm

Using the formula, V = (1/3)πr²h

Volume of cone = (1/3)π(15)²(20) = 1500π cm³

Therefore,

The volume of the cone is 1500π cm³

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