What is the Volume of a Frustum of a Cone?

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Grade Level:

Class 6

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

Definition

What is it?

A frustum of a cone is what you get when you slice off the top part of a cone parallel to its base. Its volume is the amount of space it takes up, calculated by subtracting the volume of the smaller cone (that was sliced off) from the volume of the original bigger cone.

Simple Example

Quick Example

Imagine a bucket used to carry water, like the ones we use for cleaning floors or carrying sand. If you look closely, a bucket is shaped like a frustum of a cone. Its volume tells us how much water or sand it can hold.

Worked Example

Step-by-Step

Let's find the volume of a frustum where the original cone had a radius (R) of 6 cm and height (H) of 10 cm. The top part sliced off was a smaller cone with radius (r) of 3 cm and height (h) of 5 cm.

Step 1: Calculate the volume of the original bigger cone (V_big) using the formula (1/3) * pi * R^2 * H.
V_big = (1/3) * pi * 6^2 * 10 = (1/3) * pi * 36 * 10 = 120 * pi cubic cm.
---Step 2: Calculate the volume of the smaller cone (V_small) that was sliced off using the formula (1/3) * pi * r^2 * h.
V_small = (1/3) * pi * 3^2 * 5 = (1/3) * pi * 9 * 5 = 15 * pi cubic cm.
---Step 3: Subtract the volume of the smaller cone from the volume of the bigger cone to find the frustum's volume (V_frustum).
V_frustum = V_big - V_small = 120 * pi - 15 * pi = 105 * pi cubic cm.
---Step 4: If we use pi = 3.14, then V_frustum = 105 * 3.14 = 329.7 cubic cm.
Answer: The volume of the frustum is 105 * pi cubic cm or approximately 329.7 cubic cm.

Why It Matters

Understanding volumes of shapes like frustums is crucial in engineering and design, helping engineers figure out how much material is needed for structures or how much liquid a container can hold. This skill is vital for careers in architecture, civil engineering, and even product design, ensuring things like water tanks or funnels work correctly.

Common Mistakes

MISTAKE: Students often confuse the height of the frustum with the height of the original cone or the cut-off cone. | CORRECTION: Remember, the height of the frustum (H_frustum) is the difference between the height of the original cone (H) and the height of the cut-off cone (h), i.e., H_frustum = H - h.

MISTAKE: Using the same radius for both the top and bottom circles of the frustum. | CORRECTION: A frustum has two different radii: a larger radius (R) for the base and a smaller radius (r) for the top opening.

MISTAKE: Forgetting to subtract the volume of the smaller cone from the larger cone. | CORRECTION: The volume of the frustum is specifically the 'remaining' part, so you must subtract the part that was removed.

Practice Questions

Try It Yourself

QUESTION: A cone has a base radius of 9 cm and a height of 15 cm. A smaller cone is cut off from the top, having a radius of 3 cm and a height of 5 cm. What is the volume of the frustum? (Use pi = 22/7) | ANSWER: 1232 cubic cm

QUESTION: A water cooler bottle is shaped like a frustum. Its bottom radius is 14 cm, and its top radius is 7 cm. The total height of the original cone was 30 cm, and the cut-off cone's height was 15 cm. Find the volume of the bottle. (Use pi = 22/7) | ANSWER: 6468 cubic cm

QUESTION: If the volume of a frustum is 500 * pi cubic units, and the original cone had a radius of 10 units and height of 20 units, what was the volume of the smaller cone that was cut off? | ANSWER: 166.67 * pi cubic units (approx)

MCQ

Quick Quiz

Which of these everyday objects is shaped like a frustum of a cone?

A cricket ball

A traffic cone

A water bucket

A Rubik's Cube

The Correct Answer Is:

A water bucket has a wider base and a narrower top, which is exactly the shape of a frustum. A cricket ball is a sphere, a traffic cone is a full cone, and a Rubik's Cube is a cuboid.

Real World Connection

In the Real World

From the design of a water tank in your building to the shape of a funnel used in a petrol pump, understanding frustum volumes is key. Even when engineers design parts for rockets at ISRO, calculating volumes of complex shapes helps them manage fuel capacity and weight accurately.

Key Vocabulary

Key Terms

FRUSTUM: The part of a cone that remains after the top part is cut off by a plane parallel to the base. | CONE: A 3D shape with a circular base and a single vertex. | VOLUME: The amount of space a 3D object occupies. | RADIUS: The distance from the center to the edge of a circle. | HEIGHT: The vertical distance from the base to the top.

What's Next

What to Learn Next

Great job learning about frustum volume! Next, you can explore the 'Surface Area of a Frustum of a Cone'. This will help you understand how much material is needed to cover the outside of such a shape, building on your knowledge of its 3D properties.