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How to derive the formula for voulume of frustum of a cone?

  - How to derive the relation bnetween the orginal cone radius / height to the portion of the cone that is removed?

 The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:

V = frac{h_2 B_2 - h_1 B_1}{3}

where B1 is the area of one base, B2 is the area of the other base, and h1, h2 are the perpendicular heights from the apex to the planes of the two bases.

Considering that

frac{B_1}{h_1^2}=frac{B_2}{h_2^2}

the volume can also be expressed as the product of the height h = h2−h1 of the frustum, and the Heronian mean of their areas:

V = frac{h}{3}(B_1+B_2+sqrt{B_1 B_2})

Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary no, the square root of negative one.

In particular, the volume of a circular cone frustum is

V = frac{pi h}{3}(R_1^2+R_2^2+R_1 R_2)

where π is 3.14159265..., and R1, R2 are the radii of the two bases.

Pyramidal frustum.

The volume of a pyramidal frustum whose bases are n-sided polygons is

V= frac{n h}{12} (a_1^2+a_2^2+a_1a_2)cot frac{180}{n}

where a1 and a2 are the sides of the two bases.


Surface area                                                                          

The surface area of a right circular cone frustum is

A= pileft[(R_1^2+R_2^2)+sqrt{(R_1^2-R_2^2)^2+(h(R_1+R_2))^2}right]

where R1 and R2 are the base and top radii respectively.

The surface area of a right frustum whose bases are similar regular n-sided polygon is

A= frac{n}{4}left[(a_1^2+a_2^2)cot frac{pi}{n} + sqrt{(a_1^2-a_2^2)^2sec^2 frac{pi}{n}+4 h^2(a_1+a_2)^2} right]

where a1 and a2 are the sides of the two bases.

HOPE THIS HELPS YOU....GOOD LUCK

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View Full Answer

 

Let ABC be a cone. A frustum DECB is cut by a plane parallel to its base. Let r 1 and r 2 be the radii of the ends of the frustum of the cone and h be the height of the frustum of the cone.

In ΔABG and ΔADF, DF||BG

∴ ΔABG ∼ ΔADF

CSA of frustum DECB = CSA of cone ABC − CSA cone ADE

CSA of frustum = 

 

In ΔABG and ΔADF, DF||BG

∴ ΔABG ∼ ΔADF

Volume of frustum of cone = Volume of cone ABC − Volume of cone ADE

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Prove volume of frustumof a cone
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thanks prakhar for answering this question..
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How can a frustum of cone have a volume ? Volume is only found in speakers
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The frustum as said earlier is the sliced part of a cone, therefore for calculating the volume, we find the difference of volumes of two right circular cones.

Frustum

From the figure, we have, the total height H’ = H+h and the total slant height L =l1 +l2. The radius of the cone = R and the radius of the sliced cone = r. Now the volume of the total cone = 1/3 π R2 H’ = 1/3 π R2 (H+h)

The volume of the Tip cone = 1/3 πr2h. For finding the volume of the frustum we calculate the difference between the two right circular cones, this gives us

= 1/3 π R2 H’ -1/3 πr2h
= 1/3π R2 (H+h) -1/3 πr2h
=1/3 π [ R2 (H+h)-r2 h ]

Now on seeing the whole cone with the sliced cone, we come to know that the right angle of the whole cone Δ QPS  is similar to the sliced cone Δ QAB. This gives us, R/ r = H+h / h ⇒ H+h = Rh/r. Substituting the value of H+h in the formula for the volume of frustum we get,

=1/3 π [ R2 (Rh/r)-r2 h ] =1/3 π  [R3h/r-r2 h]
=1/3 π h (R3/r-r2 )  =1/3 π h (R3-r3 / r)

The Volume of Frustum of Cone = 1/3 π h [(R3-r3)/ r]

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