What is a Frustum of a Cone?

S3-SA2-0143

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A frustum of a cone is a part of a cone that is left when a smaller cone is cut off from the top by a plane parallel to its base. Imagine slicing off the top pointy part of a cone, and what remains is a frustum. It has two circular bases of different sizes.

Simple Example

Quick Example

Think of a bucket used to carry water or sand at a construction site. This bucket has a wider opening at the top and a narrower base at the bottom, just like a frustum. Another example is a common tea glass (cutting chai glass) which is wider at the top than at the bottom.

Worked Example

Step-by-Step

Let's say we have a big cone with a height of 10 cm and a base radius of 5 cm. We cut off a smaller cone from the top, parallel to the base, at a height of 4 cm from the base. This means the smaller cone we cut off has a height of 10 - 4 = 6 cm. Its radius would be proportional to the big cone's radius. Let's find the radius of the top circular base of the frustum.

Step 1: Understand the setup. Original cone height H = 10 cm, radius R = 5 cm. The frustum is formed by cutting at 4 cm from the base, so the height of the frustum (h_frustum) = 4 cm.
---Step 2: The small cone cut off from the top has a height (h_small) = H - h_frustum = 10 cm - 4 cm = 6 cm.
---Step 3: Use similar triangles to find the radius of the top base of the frustum (which is the base of the small cone). The ratio of height to radius is the same for the big cone and the small cone. (h_small / r_small) = (H / R).
---Step 4: Substitute values: (6 / r_small) = (10 / 5).
---Step 5: Solve for r_small: 6 / r_small = 2. So, r_small = 6 / 2 = 3 cm.
---Answer: The top radius of the frustum is 3 cm, and the bottom radius is 5 cm.

Why It Matters

Understanding frustums helps engineers design everyday objects like buckets, lampshades, and even parts of rockets. In fields like Computer Science and AI/ML, these 3D shapes are used in computer graphics to render realistic objects and in robotics for path planning. Even in Physics, it helps calculate volumes of containers for experiments.

Common Mistakes

MISTAKE: Confusing a frustum with a cylinder or a full cone. | CORRECTION: Remember a frustum always has two different-sized circular bases, unlike a cylinder (two same-sized bases) or a cone (one base and a pointy top).

MISTAKE: Assuming the top and bottom radii are equal. | CORRECTION: If the radii were equal, it would be a cylinder, not a frustum. A frustum's defining feature is its tapering shape with two different radii.

MISTAKE: Incorrectly identifying the height of the frustum versus the original cone's height or the small cone's height. | CORRECTION: The height of the frustum is the perpendicular distance between its two circular bases, not the slant height.

Practice Questions

Try It Yourself

QUESTION: A bucket has a top radius of 10 cm and a bottom radius of 7 cm. Its height is 15 cm. Is this object a frustum of a cone? | ANSWER: Yes, because it has two circular bases of different sizes and a specific height.

QUESTION: If a cone with a height of 20 cm and a base radius of 8 cm is cut parallel to its base at a height of 10 cm from the base, what is the height of the frustum formed? | ANSWER: The height of the frustum is 10 cm.

QUESTION: A cone with height 30 cm and base radius 12 cm is cut by a plane parallel to the base at a height of 10 cm from the vertex. What is the radius of the top base of the frustum formed? (Hint: The height of the small cone cut off is 10 cm.) | ANSWER: The radius of the top base of the frustum is 4 cm.

MCQ

Quick Quiz

Which of these everyday objects is an example of a frustum?

A cricket ball

A water bottle (cylindrical)

A traffic cone

A bucket

The Correct Answer Is:

A bucket has a wider opening at the top and a narrower base at the bottom, which perfectly matches the shape of a frustum. A cricket ball is a sphere, a water bottle is a cylinder, and a traffic cone is a full cone.

Real World Connection

In the Real World

From the shape of an Indian 'matka' (earthen pot) to the design of some modern building pillars, frustums are all around us. Civil engineers use this concept to calculate the volume of concrete needed for structures like foundations or retaining walls that taper. Even the shape of some traditional 'diyas' (oil lamps) can be a frustum.

Key Vocabulary

Key Terms

FRUSTUM: The part of a cone remaining after a smaller cone is cut off from the top by a plane parallel to its base. | CONE: A 3D geometric shape that tapers smoothly from a flat circular base to a point called the apex. | RADIUS: The distance from the center of a circle to any point on its circumference. | HEIGHT: The perpendicular distance between the two bases of a frustum or the base and apex of a cone. | BASE: The bottom (or top) circular face of a frustum or cone.

What's Next

What to Learn Next

Great job understanding what a frustum is! Next, you can learn about calculating the volume and surface area of a frustum. This will help you find out how much water a bucket can hold or how much paint is needed for a lampshade, using the formulas we will explore.