Frustum Of Cone Definition Formula and Examples

The concept of frustum of cone formula plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Students encounter it in surface area, volume problems, and competitive exams like JEE/NEET.

What Is Frustum of Cone Formula?

A frustum of a cone is a three-dimensional shape formed when a right circular cone is sliced by a plane parallel to its base and the top portion is removed. The remaining solid is called the frustum. You’ll find this concept applied in areas such as calculating the volume of conical buckets, designing lampshades (truncated cones), and determining the quantity of materials needed for frustum-shaped machine parts.

Key Formula for Frustum of Cone

Here’s the standard frustum of cone formula for volume, curved surface area (CSA), and total surface area (TSA):

Variable meanings:
R = radius of lower (bigger) base
r = radius of upper (smaller) base
h = vertical (perpendicular) height
l = slant height (use Pythagoras, l = √[(R – r)² + h²])

Cross-Disciplinary Usage

Frustum of cone formula is not only useful in Maths but also plays an important role in Physics (e.g., calculating fluid volumes), Computer Science (3D graphics), and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in geometry, mensuration, and real-world modeling questions.

Step-by-Step Illustration

1. Start with a right circular cone. Cut it by a plane parallel to its circular base, removing the top portion. The section between the two parallel faces is called the frustum.
2. Given values:
   R = base radius
   r = top radius
   h = perpendicular height between bases
   l = slant height (if not given, compute l = √[(R – r)² + h²])
3. To find volume:
   Apply V = (1/3)πh(R² + r² + Rr)
4. To find Curved Surface Area (CSA):
   Compute CSA = π(R + r)l
5. To find Total Surface Area (TSA):
   Calculate TSA = π(R + r)l + πR² + πr²

Solved Example Problems

Problem 1 – Volume:
A frustum-shaped bucket has base radius R = 20 cm, top radius r = 10 cm, and height h = 25 cm. Find its volume. (Use π = 3.14)

1. Write the volume formula: V = (1/3)πh(R² + r² + Rr)

2. Substitute values: V = (1/3) × 3.14 × 25 × (20² + 10² + 20×10)

3. Compute inside bracket: 400 + 100 + 200 = 700

4. V = (1/3) × 3.14 × 25 × 700

5. 3.14 × 25 × 700 = 54,950

6. Divide by 3: 54,950 / 3 ≈ 18,316.67 cm³

Final Answer: The volume is approximately 18,317 cm³

Problem 2 – Surface Area:
Given a frustum with R = 7 cm, r = 3 cm, height h = 4 cm. Find CSA and TSA. (π = 22/7)

1. First calculate slant height:
l = √[(R – r)² + h²] = √[(7 – 3)² + 4²] = √[16 + 16] = √32 ≈ 5.66 cm

2. CSA = π(R + r)l = (22/7) × (7 + 3) × 5.66 = (22/7) × 10 × 5.66 ≈ 22 × 10 × 5.66 / 7 ≈ 1,245.2 / 7 ≈ 177.89 cm²

3. Area of both bases:
Lower base = πR² = (22/7) × 49 = 154 cm²
Upper base = πr² = (22/7) × 9 = 28.29 cm²
4. TSA = CSA + Area of both bases = 177.89 + 154 + 28.29 = 360.18 cm²

Final Answers: CSA ≈ 177.89 cm², TSA ≈ 360.18 cm²

Speed Trick or Vedic Shortcut

Here’s a quick trick for frustum surface area calculations during exams:

1. If slant height is not given, don’t waste time. Use l = √[(R – r)² + h²] immediately before calculating CSA. Remember, if one base radius is much larger, focus your working on the largest term in the volume formula to quickly estimate the answer’s rough scale.
2. Tip: Break (R² + r² + Rr) as (R + r)² – Rr for easy squaring: (20 + 10)² – (20×10) = 900 – 200 = 700 (this matches the first example).

Tricks like this save time — especially in competitive exams! Vedantu’s live classes offer many more such quick solutions.

Try These Yourself

• Find the volume of a frustum where R = 15 cm, r = 7 cm, h = 20 cm.
• If CSA of a frustum is 440 cm², R = 7 cm, r = 3 cm, find the slant height.
• A conical lamp shade has R = 11 cm, r = 4 cm, height h = 8 cm. Find its TSA.

Frequent Errors and Misunderstandings

• Confusing TSA with CSA (remember, TSA includes both circular bases—CSA does not!)
• Interchanging R and r between top and bottom bases.
• Forgetting to square the radii in the volume formula.
• Using slant height in the wrong place—always use 'l' for CSA!
• Units mismatch: volume is always in cubic units, area in square units.

Relation to Other Concepts

The frustum of cone formula connects closely with the cone, surface area of cone, and cylinder. Mastering this formula also helps for questions on volume of a cylinder and more complex mensuration problems in higher classes.

Classroom Tip

A quick way to remember the frustum of cone formula is to imagine cutting a cake-cone horizontally: the “frustum” is the part that remains between the two bases. Visualization — and color-coded diagrams — make the concepts click faster. Vedantu’s teachers use everyday examples like buckets, glasses, and lampshades in live classes to make frustum problems relatable.

We explored frustum of cone formula—from definition, formulas, stepwise solved problems, mistakes, and related shapes. Keep practicing with Vedantu to become fast and confident with frustum questions in your exams!

Useful related reading:

• Cone – for parent shape basics
• Surface Area of Cone
• Volume of Cube, Cuboid, and Cylinder
• Mensuration