Truncated Cone Volume Calculator

Base Radius \( R \):

Top Radius \( r \):

Height \( h \):

Volume

Volume (cm³): Volume (m³): Volume (in³): Volume (ft³): Volume (liters):

Slant Height \( s \)

Slant Height (cm): Slant Height (m): Slant Height (in): Slant Height (ft):

1. What is a Truncated Cone Volume Calculator?

Definition: This calculator computes the volume and slant height of a truncated cone (frustum), which is a cone whose top is cut off by a plane parallel to its base.

Purpose: Useful in geometry, engineering, and design for determining the volume and slant height of truncated conical shapes, such as buckets or lampshades.

2. How Does the Calculator Work?

The calculations are based on the following formulas:

\[ V = \frac{1}{3} \times \pi \times h \times (r^2 + r \times R + R^2) \] \[ s = \sqrt{(R - r)^2 + h^2} \]

Unit Conversions:

Input (Length): mm (×0.1), cm (×1), m (×100), in (×2.54), ft (×30.48) to cm
Volume (from cm³): m³ (×0.000001), in³ (×0.0610237), ft³ (×0.0000353147), liters (×0.001)
Slant Height (from cm): m (×0.01), in (×0.393701), ft (×0.0328084)

Explanation: All dimensions are converted to cm, the volume and slant height are calculated, and then converted to other units.

3. Importance of Truncated Cone Volume and Slant Height Calculation

Details: Calculating the volume and slant height of a truncated cone is essential in engineering (e.g., designing containers, funnels), manufacturing, and architecture (e.g., determining material requirements for conical shapes).

4. Using the Calculator

Tips: Enter the base radius \( R \), top radius \( r \), and height \( h \) in mm, cm, m, in, or ft (\( R > 0 \), \( r \geq 0 \), \( h > 0 \)). The result shows the volume in cm³, m³, in³, ft³, and liters, and the slant height in cm, m, in, and ft.

5. Example

Given: Base radius \( R = 6 \, \text{cm} \), top radius \( r = 2 \, \text{cm} \), height \( h = 8 \, \text{cm} \).
Volume Calculation: \( V = \frac{1}{3} \times \pi \times 8 \times (2^2 + 2 \times 6 + 6^2) = \frac{1}{3} \times \pi \times 8 \times (4 + 12 + 36) = \frac{1}{3} \times \pi \times 8 \times 52 \approx 435.839 \, \text{cm}^3 \).
Slant Height Calculation: \( s = \sqrt{(6 - 2)^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.944 \, \text{cm} \).
Conversions (Volume):
- \( \text{cm}^3 \approx 435.839 \),
- \( \text{m}^3 \approx 0.000436 \),
- \( \text{in}^3 \approx 26.598 \),
- \( \text{ft}^3 \approx 0.0154 \),
- \( \text{liters} \approx 0.436 \).
Conversions (Slant Height):
- \( \text{cm} \approx 8.944 \),
- \( \text{m} \approx 0.089 \),
- \( \text{in} \approx 3.521 \),
- \( \text{ft} \approx 0.293 \).

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