Surface Areas and Volumes

Frustum of cone

Drinking water is the most common activity. You must be wondering what does water have to do with volume, well do you know how much water do you drink in a day? Have you ever calculated the volume of the glass in which you drink water. Glass is an example of the frustum of a cone. In the chapter below we shall help you calculate the volume of a glass.

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Frustum of a Cone

We have already learned in our previous classes to find the volume of perfectly shaped 3-d structures. Cones, Cylinders etc have been under our consideration for finding the respective volumes. These are some of the general shapes! In this chapter, we shall deal with shapes that we use the most. Look at the picture of a glass given below. What do you find different in a glass tumbler? Which regular shape does it resemble? A cone or a cylinder or none.

Frustum

Source: Flickr

A glass tumbler resembles a cone with it’s pointed part sliced. Yes, a glass is a resemblance of the frustum of a cone. When the smaller end, parallel to the base is cut from a cone, the shape we get is called the frustum of the cone. The case of measurement here is different. So, how do we calculate the volume of the frustum now?

Browse more Topics under Surface Areas And Volumes

Learn more about Cuboid and Cube here in detail here.

In the combination of solids, we added the volumes of two adjoining shapes which gave us the total volume of any structure. But for frustum of the cone as we are slicing the smaller end of the cone as shown in the figure, hence we need to subtract the volume of the sliced part.

Frustum

The Volume of the Frustum of a Cone

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 R(H+h) -1/3 πr2h
=1/3 π [ R(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 [ R(Rh/r)-r2 h ] =1/3 π  [R3h/r-r2 h]
=1/3 π h (R3/r-r)  =1/3 π h (R3-r/ r)

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

Similar Triangles Property

Using the same Similar triangles property lets find the value of h, R/ r = (H+h)/ h.
⇒ h= [r/(R-r)] H. Substituting the value of h this equation we get: =1/3 πH [r/(R-r)][(R3-r3)/ r)\]
=1/3 πH [(R3-r3)/(R-r)]
= 1/πH [(R-r)(R+Rr+r2)/ (R-r) ]
=1/πH (R+Rr+r2)
Therefore, the volume (V) of the frustum of the cone is =1/3 πH (R+Rr+r).

Learn more about Area and Volume of Combination of Solids here

Curved Surface Area and Total Surface Area of the Frustum

The curved surface area of the frustum of the cone = π(R+r)l

The total surface area of the frustum of the cone = π l(R+r) +πR+πr2

The slant height (l1) in both the cases shall be  = √[H2 +(R-r)2]

These equations have been derived using the similarity of triangles property between the two triangles QPS and QAB. Measurement of the volume, surface area and curved surface area like any other measurement depends on the understanding of the subject.

For the combination of solids, we add all the constituting shapes. Here since we are slicing a similar triangle from the cone, we find the difference between the two shapes. These two parts of the measurements involve operations and depend highly on the logic of understanding.

Sample Question for You

Question 1: An open plastic drum of height 63 cm with radii of lower and upper ends as 15 cm and 25 cm respectively is filled with Milk. Find the cost of milk which can completely fill the bucket at Rs. 45 per litre. Also find the surface area of the drum, if it needs to be coloured with at the rate of 50 ps/sq cm.

Answer : A plastic drum resembles a frustum of a cone. The Height (h) of the drum=63 cm. Upper Radius (R) of the drum = 25 cm. Lower Radius (r) of the drum= 15 cm. Using the formula for the volume of the frustum of the cone the Volume (V) of the drum shall be: 1/3πH (R+ Rr + r2)

= 1/3 × 22/7 × 63(252  + 25×15 +152)
= 66 (625 + 375 + 225)
= 80,850 cm3
= 80 L 850 ml
One litre of milk costs Rs 45, and 80 L & 850 ml shall cost = 80.850×45 = Rs 3,638.25

The Drum stores milk of cost = Rs 3,638.25. Total Surface Area of the Drum = π l (R+r) + πR2  + πr2
l =√H+(R-r)2
l = 63+ (25-15) 2
= √3969+100
= 63.79 cm

Using the formula πl (R+r) +πR+πr2
=22/7 × 63.79 (25+15) + 22/7 (25)+ 22/7(15)2
= 22/7 × 63.79 × 40 + 22/7 × 625 + 22/7 × 225
= 22/7[2551.6+625+225] = 10,690.74 cm2
Cost of painting per sq cm is 50 ps.
So for 10,690.74 sq cm the cost of painting shall be 10,690.74 × 50 = Rs 5,345 approximately.

Question 2: What is the formula for the frustum?

Answer: The conical Frustum formulas in terms of r and h is as follows:

Volume of a conical frustum = V = (1/3) * π * h * (r12 + r22 + (r1 * r2)).

Question 3: What is the volume of a frustum?

Answer: 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].

Question 4: What is Curved Surface Area (CSA) of a frustum?

Answer: The Curved Surface Area (CSA) of the frustum of a cone is: = pi * l(R + r) where the (r) stands for = radius of the smaller circle and (R) stands for = radius of the bigger circle and the (l) = slant height of the frustum.

Question 5: Is frustum a polyhedron?

Answer: The polyhedron got by cutting a pyramid with a plane that is parallel to its base and meets all of its generatrices. From the 2 polyhedrons obtained, the polyhedron that does not comprise the apex of the pyramid is known as the frustum of the pyramid.

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