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?
A frustum of a cone is a three-dimensional geometric shape formed by cutting off the top of a cone with a plane parallel to its base. It has two circular bases of different sizes and a curved lateral surface. Think of it like a truncated cone or a cone with its top cut off. Frustums are common in everyday objects, such as lampshades, buckets, and certain machine parts.
2. What is the formula for the volume of a frustum of a cone?
The volume (V) of a frustum of a cone is calculated using the formula: V = (1/3)πh(R² + r² + Rr), where h is the height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base. This formula is derived by subtracting the volume of a smaller cone from a larger cone.
3. How do you calculate the curved surface area (CSA) of a frustum?
The curved surface area (CSA) of a frustum is given by the formula: CSA = πl(R + r), where l is the slant height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base. The slant height (l) can be calculated using the Pythagorean theorem: l = √(h² + (R - r)²), where h is the height of the frustum.
4. What is the formula for the total surface area (TSA) of a frustum of a cone?
The total surface area (TSA) of a frustum is the sum of its curved surface area and the areas of its two circular bases. The formula is: TSA = πl(R + r) + πR² + πr², where l is the slant height, R is the radius of the larger base, and r is the radius of the smaller base.
5. What are the units used in frustum formulas?
The units used depend on the units of measurement for the radius and height. If radius is in centimeters (cm) and height in centimeters (cm), then volume will be in cubic centimeters (cm³), and surface area will be in square centimeters (cm²). Always maintain consistent units throughout your calculations to avoid errors.
6. How is the frustum formula derived?
The frustum volume formula is derived by considering the frustum as the difference between two cones. By using similar triangles and the formula for the volume of a cone, we can derive the frustum's volume formula. A similar approach using the surface area of cones leads to the frustum's surface area formulas. Detailed derivations can be found in most geometry textbooks.
7. What are some common mistakes students make when using frustum formulas?
Common mistakes include: Using incorrect formulas, mixing up the radii (R and r), forgetting to calculate the slant height (l), and using inconsistent units. Carefully check your values and units before performing calculations. Always double-check your work after solving.
8. How can I visualize the difference between a cone and a frustum?
Imagine a cone. Now, imagine slicing it horizontally to remove the top portion. The remaining bottom part is the frustum. The key difference is that a cone has one circular base and a vertex, while a frustum has two parallel circular bases of unequal size.
9. What are some real-world applications of frustum calculations?
Frustum calculations are used in various fields. Examples include determining the volume of containers with tapered shapes (buckets, funnels), calculating the material needed for construction (lampshades, roofs), and in engineering design (machine parts, storage tanks).
10. Can I use a calculator to solve frustum problems?
Yes, using a calculator (or online calculator) can significantly speed up calculations, particularly for complex problems. Many online calculators are specifically designed for frustum calculations, simplifying the process.
11. What is the difference between a frustum and a truncated cone?
The terms "frustum" and "truncated cone" are often used interchangeably. Both refer to the three-dimensional shape formed when a cone is cut by a plane parallel to its base. There's no practical difference in their geometric meaning.
