Frustum of a Cone

Definition: A frustum of a cone is the portion of a cone that lies between its base and a plane cutting the cone parallel to the base.

Key elements of a frustum:

• R = radius of the larger circular face (base of the original cone)
• r = radius of the smaller circular face (cut face)
• h = height of the frustum (perpendicular distance between the two circular faces)
• l = slant height of the frustum

Relationship between slant height, height, and radii:

l = sqrt[h^2 + (R - r)^2]

How a frustum is formed:

1. Take a right circular cone with base radius R and height H.
2. Cut it with a plane parallel to the base at height h₁ from the base (so h₁ < H).
3. Remove the smaller cone from the top.
4. The remaining solid is the frustum with larger radius R, smaller radius r, and height h₁.

Derivation of Volume Formula:

A frustum is a large cone minus a small cone cut from the top.

1. Let the large cone have height H and base radius R.
2. Let the small cone (removed) have height H - h and base radius r.
3. By similar triangles: r/R = (H - h)/H, which gives H = Rh/(R - r).
4. Volume of large cone = (1/3)πR^2H = (1/3)πR^2 × Rh/(R - r) = (1/3)πR^3h/(R - r).
5. Volume of small cone = (1/3)πr^2(H - h) = (1/3)πr^2 × rh/(R - r) = (1/3)πr^3h/(R - r).
6. Volume of frustum = (1/3)πh/(R - r) × (R^3 - r^3).
7. Using factorisation: R^3 - r^3 = (R - r)(R^2 + r^2 + Rr).
8. Volume = (1/3)πh × (R^2 + r^2 + Rr). Hence derived.

Derivation of CSA Formula:

1. CSA of large cone = πRL₁ (where L₁ = slant height of large cone).
2. CSA of small cone = πrL₂ (where L₂ = slant height of small cone).
3. CSA of frustum = πRL₁ - πrL₂.
4. By similar triangles: L₂/L₁ = r/R, so L₂ = rL₁/R.
5. CSA = πRL₁ - πr(rL₁/R) = πL₁(R^2 - r^2)/R.
6. Also, l = L₁ - L₂ = L₁(1 - r/R) = L₁(R - r)/R, so L₁ = lR/(R - r).
7. CSA = π × lR/(R - r) × (R^2 - r^2)/R = πl(R + r)(R - r)/(R - r) = π(R + r)l.

Derivation of Slant Height:

• Drop a perpendicular from the edge of the smaller circle to the larger base.
• This forms a right triangle with legs h (height) and (R - r) (difference of radii).
• By Pythagoras: l^2 = h^2 + (R - r)^2.

Problems on frustum of a cone in Class 10 include:

Type 1: Volume of a Frustum

• Given R, r, and h, find the volume using V = (1/3)πh(R^2 + r^2 + Rr).

Type 2: CSA and TSA

• Find slant height first using l = sqrt[h^2 + (R - r)^2], then compute CSA and TSA.

Type 3: Real-World Objects (Bucket, Flowerpot, Lampshade)

• Given dimensions of a bucket-shaped object, find volume (capacity) or surface area (material needed).

Type 4: Metal Conversion Problems

• A frustum-shaped container is melted and recast into another shape (cylinder, sphere, etc.). Find the dimensions of the new shape using volume conservation.

Type 5: Finding Missing Dimensions

• Given volume and some dimensions, find the missing radius or height.

Type 6: Cost Problems

• Find the cost of painting the outer surface (TSA) or capacity (volume) at a given rate.

Step-by-step method for frustum problems:

1. Identify the shape — confirm it is a frustum (two parallel circular faces of different radii).
2. List the given values — R (larger radius), r (smaller radius), h (height), l (slant height).
3. Find slant height if needed — l = sqrt[h^2 + (R - r)^2].
4. Apply the correct formula:
   • Volume: (1/3)πh(R^2 + r^2 + Rr)
   • CSA: π(R + r)l
   • TSA: π(R + r)l + πR^2 + πr^2
5. Substitute and calculate — use π = 22/7 or 3.14 as given in the problem.

Common Mistakes to Avoid:

• Confusing R and r — R is always the LARGER radius.
• Using l (slant height) where h (vertical height) is needed, or vice versa.
• Forgetting the (R - r) in the slant height formula (not R + r).
• In TSA, forgetting to add the areas of both circular faces (πR^2 + πr^2).
• In bucket problems, the open top means TSA = CSA + πR^2 (only the bottom circle, not both).

Household Items:

• Buckets, flower pots, lamp shades, and drinking glasses are often frustum-shaped. Calculating capacity and material required uses frustum formulas.

Civil Engineering:

• Foundations of buildings and dams often have frustum-shaped cross-sections for structural stability.

Manufacturing:

• Metal casting frequently involves frustum-shaped moulds. Knowing the volume helps determine the amount of raw material needed.

Agriculture:

• Water tanks, grain hoppers, and feed containers are often frustum-shaped for ease of emptying from the bottom.

Architecture:

• Cooling towers of power plants are hyperboloid structures that approximate frustums in cross-section.