Truncated Cone Calculator
Calculate volume, surface area, and properties of truncated cones (frustums)
Calculate Truncated Cone Properties
Truncated Cone Properties
Calculation Details:
Volume Formula: V = (1/3) × π × h × (r² + r × R + R²)
Slant Height: s = √((R - r)² + h²)
Lateral Area: π × (R + r) × s
Total Area: Lateral + Top + Bottom areas
Truncated Cone Diagram
R: Base radius (larger)
r: Top radius (smaller)
h: Height (perpendicular)
s: Slant height
Key Formulas
Volume:
V = (1/3) × π × h × (r² + r×R + R²)
Slant Height:
s = √((R - r)² + h²)
Lateral Area:
A_lateral = π × (R + r) × s
Surface Areas:
Top = π × r²
Bottom = π × R²
Total = Lateral + Top + Bottom
Quick Tips
Base radius must be larger than top radius
Also known as a frustum of a cone
Common in lamp shades, buckets, and containers
Slant height is always longer than height
Understanding Truncated Cones
What is a Truncated Cone?
A truncated cone, also called a frustum, is a cone with its apex (top point) cut off by a plane parallel to the base. This creates a 3D shape with two circular faces of different sizes connected by a curved lateral surface.
Truncated cones are common in everyday objects like lamp shades, buckets, traffic cones, and many containers. Understanding their properties is essential for engineering, architecture, and manufacturing applications.
Key Properties
Volume: The amount of space inside the truncated cone
Lateral Surface Area: The curved surface connecting the two circles
Total Surface Area: Lateral area plus both circular faces
Slant Height: Distance along the lateral surface between the circles
Mathematical Relationships
Volume Derivation
The volume formula comes from subtracting a smaller cone from a larger cone:
V = (1/3)πh(r² + rR + R²)
Slant Height Formula
Uses the Pythagorean theorem with height and radius difference:
s = √((R - r)² + h²)
Surface Area
Combines the lateral surface with top and bottom circles:
A = π(R + r)s + πr² + πR²
Real-World Applications
Engineering & Construction
Calculate material volumes for concrete foundations, storage tanks, hoppers, and structural elements with tapered designs.
Manufacturing & Design
Design lamp shades, containers, funnels, and other products requiring precise volume and surface area calculations.
Food & Beverage Industry
Calculate capacity of tapered containers, cups, bottles, and packaging with truncated cone shapes.