Truncated Cone Volume Calculator
Calculate the volume of truncated cones (frustums) with precision
Calculate Truncated Cone Volume
Volume Calculation Results
Calculation Details:
Method: Direct calculation using radii and height
Formula Used: V = (1/3) × π × h × (r² + r × R + R²)
Height: 4 cm
Step-by-step:
- r² = 9
- R² = 25
- r × R = 15
- r² + r×R + R² = 49
- V = (1/3) × π × 4 × 49 = 205.2507 cm³
Volume Visualization
V: Volume of the frustum
r: Top radius (smaller)
R: Base radius (larger)
h: Height (perpendicular)
Volume Formula
Main Formula:
V = (1/3) × π × h × (r² + r×R + R²)
Where:
V: Volume of truncated cone
π: Pi (≈ 3.14159)
h: Height of frustum
r: Top radius
R: Base radius
Derivation:
Volume = Large cone - Small cone
After mathematical simplification
Quick Examples
Example 1:
r = 2cm, R = 5cm, h = 6cm
V = 244.92 cm³
Example 2:
r = 1cm, R = 3cm, h = 4cm
V = 61.26 cm³
Tip:
Always ensure R > r for a valid frustum
Understanding Truncated Cone Volume
Volume Formula Derivation
The truncated cone volume formula comes from a clever geometric insight: imagine a complete cone, then subtract the volume of the smaller cone that was "cut off" to create the frustum.
Through mathematical analysis involving similar triangles and integration, this leads to the elegant formula: V = (1/3) × π × h × (r² + r×R + R²).
Why This Formula Works
• r²: Contributes the top circular area component
• R²: Contributes the bottom circular area component
• r×R: Accounts for the linear transition between circles
• (1/3)×π×h: Standard volume factor for cone-like shapes
Calculation Methods
Method 1: Radii + Height
Most common method when you know both radii and the perpendicular height.
Direct formula application
Method 2: Radii + Slant Height
When slant height is known, calculate perpendicular height first using Pythagorean theorem.
h = √(s² - (R-r)²)
Method 3: Diameters + Height
Convert diameters to radii (divide by 2), then apply standard formula.
r = d/2, R = D/2
Real-World Volume Applications
Storage & Containers
Calculate capacity of tapered tanks, silos, hoppers, and storage vessels for accurate inventory and flow management.
Manufacturing & Design
Determine material requirements for lampshades, funnels, cups, and other products with truncated cone shapes.
Engineering & Construction
Calculate concrete volumes for foundations, calculate material needs for structural elements with tapered designs.