To find the volume of a truncated cone, also known as a frustum, you use a specific geometric formula that accounts for its two different-sized circular bases and its height.
The formula to calculate the volume of a partial cone is given as:
Volume of a partial cone, V = 1/3 × πh(R² + Rr + r²)
Where:
- V represents the volume of the truncated cone.
- π (Pi) is a mathematical constant approximately equal to 3.14159.
- h is the perpendicular height of the frustum (the distance between the two bases).
- R is the radius of the larger base.
- r is the radius of the smaller base (such that R ≥ r).
This formula is derived from subtracting the volume of the smaller cone (that was 'cut off' from the top) from the volume of the original larger cone.
Understanding the Components of a Truncated Cone (Frustum)
A truncated cone is essentially what's left of a cone after its top is removed by a plane cut parallel to the base.
Here's a breakdown of the terms used in the volume formula:
| Term | Description | Unit (Example) |
|---|---|---|
| V | Volume of the truncated cone (frustum) | cubic units |
| π | Mathematical constant (approx. 3.14159) | N/A |
| h | Perpendicular height between the two circular bases | linear units |
| R | Radius of the larger circular base | linear units |
| r | Radius of the smaller circular base | linear units |
Note: Linear units could be centimeters (cm), meters (m), inches (in), etc. Cubic units would then be cm³, m³, in³, etc.
Steps to Calculate the Volume
Using the formula V = 1/3 × πh(R² + Rr + r²), you can calculate the volume by following these steps:
- Identify the height (h): Measure or find the perpendicular distance between the center of the two circular bases.
- Identify the radii (R and r): Measure or find the radius of the larger base (R) and the radius of the smaller base (r). Ensure R ≥ r. If given diameters, divide them by 2 to get the radii.
- Substitute the values: Plug the values of h, R, and r into the formula.
- Calculate R² and r²: Square the values of the large radius (R) and the small radius (r).
- Calculate the Rr term: Multiply the large radius (R) by the small radius (r).
- Sum the terms in the parenthesis: Add R², Rr, and r².
- Multiply by πh: Multiply the sum from step 6 by π and then by the height (h).
- Multiply by 1/3: Finally, divide the result from step 7 by 3 (or multiply by 1/3) to get the volume.
Example Calculation
Let's calculate the volume of a truncated cone with the following dimensions:
- Height (h) = 10 cm
- Radius of larger base (R) = 6 cm
- Radius of smaller base (r) = 4 cm
- Formula: V = 1/3 × πh(R² + Rr + r²)
- Substitute values: V = 1/3 × π × 10 × (6² + (6 × 4) + 4²)
- Calculate squares and Rr: V = 1/3 × π × 10 × (36 + 24 + 16)
- Sum terms in parenthesis: V = 1/3 × π × 10 × (76)
- Multiply by πh: V = 1/3 × π × 760
- Final Calculation: V = (760π) / 3 cm³
Using the approximate value of π ≈ 3.14159:
V ≈ (760 × 3.14159) / 3 cm³
V ≈ 2387.6084 / 3 cm³
V ≈ 795.869 cm³
So, the volume of the truncated cone is approximately 795.87 cubic centimeters.
