Mensuration • Topic 2 of 3

Frustum of a Cone

What is a frustum of a cone? If a right circular cone is cut cleanly by a flat plane parallel to its base, and the small cone at the top is removed, the remaining lower slice of the cone is called a frustum of a cone.

Think of everyday utility objects like a bucket of water, a glass tumbler, a Turkish coffee pot, or a lampshade. These objects are wide at one end and narrow at the other, featuring two flat circular faces of different sizes.

A frustum is defined by four core measurements:

  • Height (h): The vertical straight distance between the top and bottom circular bases.
  • Slant Height (l): The slanted edge distance along the outer side wall.
  • Large Radius (R): The radius of the larger circular base face.
  • Small Radius (r): The radius of the smaller circular base face.

Formulas for a frustum of a cone:

  • Slant Height (l) = $\sqrt{h^2 + (R - r)^2}$
  • Curved Surface Area (CSA) = $\pi \cdot (R + r) \cdot l$
  • Total Surface Area (TSA) = $\pi \cdot (R + r) \cdot l + \pi \cdot R^2 + \pi \cdot r^2$
  • Volume = $(1/3) \cdot \pi \cdot h \cdot (R^2 + r^2 + R \cdot r)$

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DIAGRAM 1: FORMATION OF A FRUSTUM FROM A FULL CONE

                       /\
                      /  \   [ Small Removed Cone ]
                     /____\
                    ------------ <-- Cutting Plane Parallel to Base
                   /        \
                  /          \
                 /  Frustum   \
                /   of Cone    \
               /________________\
                   Base Face

DIAGRAM 2: ANATOMY OF A FRUSTUM OF A CONE

                     . - ~ r ~ - .     <-- Small Base Radius (r)
                   /       |       \
                  /________|________\
                 /         |         \
                /          | h        \  Slant Height (l)
               /           |           \
              /____________|____________\
              . - - - - -  O  - - - - - .
                    <------ R ------>  <-- Large Base Radius (R)
Solved Examples
4
Worked Example
A metal bucket is in the shape of a frustum of a cone. The radii of its top and bottom circular ends are 20 cm and 8 cm respectively, and its vertical height is 16 cm. Find the slant height of the bucket.
Solution
  1. Step 1: Extract the given metric dimensions.
  2. Large Radius ($R$) = 20 cm.
  3. Small Radius ($r$) = 8 cm.
  4. Vertical Height ($h$) = 16 cm.
  5. Step 2: Use the slant height formula for a frustum.
  6. $$l = \sqrt{h^2 + (R - r)^2}$$
  7. Step 3: Substitute the dimensions into the equation.
  8. $$l = \sqrt{16^2 + (20 - 8)^2}$$
  9. $$l = \sqrt{25^2 - 9^2 + \dots} = \sqrt{16^2 + 12^2}$$
  10. $$l = \sqrt{256 + 144} = \sqrt{400}$$
  11. Step 4: Solve the square root.
  12. $l = 20$ cm.
  13. Answer: The slant height of the bucket is 20 cm.
5
Worked Example
A drinking glass tumbler is shaped like a frustum of a cone with a height of 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the total internal capacity (volume) of the glass. (Take $\pi = 22/7$).
Solution
  1. Step 1: Convert diameters to radii.
  2. Large Radius ($R$) = $4 / 2 = 2$ cm.
  3. Small Radius ($r$) = $2 / 2 = 1$ cm.
  4. Height ($h$) = 14 cm.
  5. Step 2: Apply the volume formula for a frustum.
  6. Volume = $(1/3) \cdot \pi \cdot h \cdot (R^2 + r^2 + R \cdot r)$
  7. Step 3: Put the numbers into the formula.
  8. Volume = $(1/3) \cdot (22/7) \cdot 14 \cdot (2^2 + 1^2 + 2 \cdot 1)$
  9. Volume = $(1/3) \cdot 22 \cdot 2 \cdot (4 + 1 + 2)$
  10. Volume = $(1/3) \cdot 44 \cdot 7$
  11. Step 4: Calculate the final value.
  12. Volume = $308 / 3 = 102.67$ cubic cm.
  13. Answer: The capacity of the glass is 308/3 cubic cm (or approximately 102.67 cubic cm).
6
Worked Example
A decorative lampshade shaped as a frustum of a cone has a small upper radius of 6 cm, a large lower radius of 14 cm, and a slant height of 10 cm. Calculate the cost of fabric needed to cover its outer curved walls if the fabric costs 5 dollars per square centimeter. (Take $\pi = 22/7$).
Solution
  1. Step 1: Identify the required area type.
  2. A lampshade is hollow at the top and bottom. We only need the Curved Surface Area (CSA).
  3. Step 2: Apply the curved surface area formula.
  4. CSA = $\pi \cdot (R + r) \cdot l$
  5. Step 3: Substitute the known values.
  6. CSA = $(22/7) \cdot (14 + 6) \cdot 10$
  7. CSA = $(22/7) \cdot 20 \cdot 10 = 4400 / 7$ square cm.
  8. Step 4: Calculate the total cost based on the area.
  9. Cost = Area $\cdot$ Rate = $(4400 / 7) \cdot 5 = 22000 / 7 = 3142.85$ dollars.
  10. Answer: The total cost of the fabric is 22000/7 dollars (or approximately 3142.85 dollars).
  11. --

Key Points

  • A frustum is formed by cutting a cone parallel to its base and removing the top vertex portion.
  • It features two separate circular base faces with different radii, labeled $R$ (large) and $r$ (small).
  • The slant height is calculated as $l = \sqrt{h^2 + (R - r)^2}$.
  • The curved wall area depends on both radii: $\text{CSA} = \pi(R + r)l$.
  • The volume formula modifies the standard cone formula to include both bases: $V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)$.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A frustum is formed by cutting a cone with a plane:
Explanation: Parallel cut removes the top cone.
Q2.The volume of a frustum is:
Explanation: Standard frustum volume.
Q3.The slant height of a frustum is:
Explanation: From the right triangle.
Q4.The CSA of a frustum is:
Explanation: $\pi(R+r)l$.
Practice more MCQs → 📝 Assignment · 35 marks