Frustum Volume
Volume of a truncated cone.
Formula
Variables
Example
A bucket shape
How it works
This formula gives the volume of a cone-like solid that has been cut by a plane parallel to its base, leaving two circular faces of different radii. It works because a frustum can be viewed as the difference between two similar cones, and the terms R^2, Rr, and r^2 come from the similarity relationship that remains after subtracting the smaller cone from the larger one. The factor 1/3\pi h is the same scaling that appears in the volume of any cone-shaped solid, with the mixed term accounting for the taper between the two ends.
When to use it
Use this when you need the capacity or material volume of tapered containers, structural supports, funnels, lamp shades, or any solid shaped like a cone with the tip removed. It is also common in geometry problems where you are given the two circular radii and the perpendicular height.
Worked example, step by step
A concrete barrier section is shaped like a frustum with bottom radius R = 6 cm, top radius r = 3 cm, and height h = 10 cm. Step 1: Write the formula: V = (1/3)\pi h(R^2 + Rr + r^2) Step 2: Substitute the values: V = (1/3)\pi(10)(6^2 + 6\cdot3 + 3^2) Step 3: Square and multiply: V = (10/3)\pi(36 + 18 + 9) Step 4: Add inside the brackets: V = (10/3)\pi(63) Step 5: Simplify: V = 210\pi cm^3 Step 6: Approximate: V \approx 659.7 cm^3
Common mistakes to avoid
- Using the slanted side length instead of the perpendicular height h
- Forgetting to square both radii before adding them into the formula
- Swapping diameter for radius without dividing by 2 first
Did You Know?
A frustum is a cone with its tip sliced off — the shape of buckets, lampshades, and the Great Pyramid's theoretical top.
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