Question

a metallic sheet form of a sector of a circle of radius 21cm has central angle of 216°.The sector is made into a cone by bringing the bounding radii together. Find the volume of cone formed​

Answer 1

Answer

2794.176 cm³

Solution

A metallic sheet is form of a sector of a circle of radius 21 cm and has central angle of 216°

→ The cone formed by it would have

l (slant height) = 21 cm

And, part of circumference of the sector would be the circumference of base

Part of circumference formed by sector = 2πr × ∅/360

Taking π as 22/7,

→ 2 × 22/7 × 21 × 216/360

→ 79.2 cm

This would act as the circumference of the base of cone

Let the base radius of cone be R

→ 2πR = 79.2 cm

→ R = 79.2 × 7/22 × 1/2

→ R = 12.6 cm

Now, we know that

l² = h² + R²

So, we will find height of the cone from here

→ 21² = h² + (12.6)²

→ h² = 441 - 158.76

→ h² = 282.24

→ h = 16.8 cm

Now we know that volume of cone = 1/3 × πr²h

→ 1/3 × 22/7 × 12.6 × 12.6 × 16.8

→ 2794.176 cm³

Answer 2

Question :--- a metallic sheet form of a sector of a circle of radius 21cm has central angle of 216°.The sector is made into a cone by bringing the bounding radii together. Find the volume of cone formed ?

Formula and concept used :--

when a circular sheet is cast into cone, than :----

→ Circumference of circular sector becomes , circumference of base of cone .

→ Radius of circular sheet becomes slant height of cone.

→ Circumference of circular sheet of sector = 2πr * (Angle at centre /360°) .

→ Slant Height of cone = √(Height)² + (Base radius)²

→ Volume of cone = 1/3 * π * R² * H .

______________________________

❁❁ Refer To Image First .. ❁❁

As told above , to find radius of cone , first we need to find circumference of sector and than compare it with Base circumference of cone.

So, Given,

→ Angle at centre = 216°

→ Radius = 21cm .

→ Let radius of Base of cone = r cm.

Comparing both circumference now, we get,

→ (216/360) * 2π * 21 = 2π * r

2π will be cancel From both sides ,

→ r = (216*21)/360

→ r = 12.6cm.

____________________________

Now, From image we can see that, Slant Height of cone is 21cm.

we also have now , radius of base = 12.6cm.

So, Putting values again , now,

→ Slant Height of cone = √(Height)² + (Base radius)²

Squaring both sides

→ l² = H² + r²

→ H² = l² - r²

putting values

→ H² = (21)² - (12.6)²

→ H² = 441 - 158.76

→ H² = 282.24

Square root both sides now,

→ H = 16.8 cm. (As negative value not possible).

______________________________

Now, we have , Height of cone so formed , and its Base radius also.

So, Putting values in Volume formula now,

→ Volume of cone = 1/3 * π * r² * H

→ Volume = 1/3 * 22/7 * (12.6)² * 16.8

→ Volume = 1/3 * 22 * 12.6 * 1.8 * 16.8 (7 cancel).

→ Volume = 22 * 4.2 * 1.8 * 16.8 (3 cancel)

→ Volume = 2794.176 cm³ .

⛬ Volume of Cone so formed will be 2794.176cm³.