How to Derive and Apply the Frustum of Cone Formula Step-by-Step
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):
| Type | Formula | Variables | Units |
|---|---|---|---|
| Volume (V) | V = (1/3)πh(R² + r² + Rr) | h = height R = larger base radius r = smaller (top) base radius |
cubic units |
| Curved Surface Area (CSA) | CSA = π(R + r)l | l = slant height (l = √[(R – r)² + h²]) | square units |
| Total Surface Area (TSA) | TSA = CSA + πR² + πr² | as above | square units |
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
- 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.
- 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]) - To find volume:
Apply V = (1/3)πh(R² + r² + Rr)
- To find Curved Surface Area (CSA):
Compute CSA = π(R + r)l
- 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:
- 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.
- 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
FAQs on Frustum of Cone Formula – Volume, Surface Area, and Examples
1. What is a frustum of a cone in simple terms?
A frustum of a cone is the part of a cone that is left after its top is cut off by a plane parallel to its base. Imagine a standard ice cream cone; if you slice off the pointy tip horizontally, the remaining bucket-shaped object is a frustum. It has two circular bases (a top and a bottom) that are parallel but have different radii, and a curved surface connecting them.
2. What are the main formulas used for a frustum of a cone?
The key formulas for a frustum of a cone, where 'R' is the radius of the larger base, 'r' is the radius of the smaller base, 'h' is the vertical height, and 'l' is the slant height, are:
- Slant Height (l): l = √[h² + (R - r)²]
- Volume (V): V = (1/3)πh(R² + r² + Rr)
- Curved Surface Area (CSA): CSA = πl(R + r)
- Total Surface Area (TSA): TSA = CSA + Area of Top Base + Area of Bottom Base = πl(R + r) + πr² + πR²
3. How is the slant height of a frustum different from its vertical height?
The vertical height (h) is the perpendicular distance between the centers of the two circular bases. It measures the straight-up height of the frustum. In contrast, the slant height (l) is the diagonal distance along the sloped, curved surface of the frustum. It is always longer than the vertical height and is crucial for calculating the surface area, not the volume.
4. What are some real-world examples of a frustum of a cone?
The frustum shape is very common in everyday objects. Some important examples include:
- A bucket or a pail used for carrying water.
- A typical glass or tumbler for drinking.
- A lampshade that directs light downwards.
- A funnel used for pouring liquids into a small opening.
- The base of some traffic cones or architectural pillars.
5. How is the volume formula for a frustum derived?
The formula for the volume of a frustum is derived by considering it as a large cone with a smaller cone removed from its top. The derivation involves these key steps:
- Imagine the full cone from which the frustum was cut. Let its height be H and radius be R.
- The smaller cone that was cut off has a height (H-h) and radius r.
- Using the property of similar triangles, a relationship between H, h, R, and r is established.
- The volume of the frustum is then calculated as: (Volume of the large cone) - (Volume of the small cone).
- After substituting the relationship from similar triangles and simplifying, we arrive at the final formula: V = (1/3)πh(R² + r² + Rr).
6. What happens to the surface area formula if the frustum is open at the top, like a bucket?
If a frustum is open at the top (like a bucket or a glass), it has no top circular surface. In this case, the Total Surface Area (TSA) calculation changes. You would calculate the sum of the Curved Surface Area (CSA) and the area of only the bottom base. The formula becomes:
TSA of an open frustum = πl(R + r) + πR².
The volume formula remains the same as it measures the capacity inside the object.
7. What are the most common mistakes to avoid when solving frustum problems?
Students often make a few common errors when working with frustum formulas. The most frequent mistakes include:
- Confusing Radii: Mixing up the larger radius (R) and the smaller radius (r) in the formulas.
- Using Height for Slant Height: Using the vertical height (h) in the surface area formulas (CSA/TSA) instead of calculating and using the slant height (l).
- Calculation Errors: Making simple arithmetic mistakes, especially when calculating the (R-r)² term for the slant height or the Rr term for the volume.
- Inconsistent Units: Using different units for height and radius (e.g., height in metres and radius in centimetres) without converting them first.
8. Is there a difference between a 'frustum' and a 'truncated cone'?
No, in the context of geometry, there is no difference between a frustum of a cone and a truncated cone. The terms are used interchangeably to describe the same three-dimensional shape—a cone that has been 'truncated' or cut by a plane parallel to its base.
