Area of a Frustum of a Cone Calculator
Calculate the surface area of a frustum (truncated cone) with step-by-step explanations
Calculate Frustum Surface Area
Radius of the smaller circular base
Radius of the larger circular base
Perpendicular distance between the two bases
Surface Area Results
Common Frustum Examples
Bucket Shape
R=8, r=5, H=10
Area ≈ 628.32
Lamp Shade
R=12, r=8, H=15
Area ≈ 1,658.76
Traffic Cone
R=15, r=3, H=20
Area ≈ 1,847.26
Frustum Tips
A frustum is a truncated cone with two parallel circular bases
Base radius must be larger than top radius
Slant height = √(H² + (R-r)²)
Surface area includes both bases plus lateral surface
Common in engineering: hoppers, funnels, containers
Understanding Frustum of a Cone
What is a Frustum?
A frustum of a cone is the portion that remains when a cone is cut by two parallel planes. It has two circular bases of different sizes connected by a curved lateral surface.
Key Components
- Large Base: The larger circular base with radius R
- Small Base: The smaller circular base with radius r
- Height: Perpendicular distance between bases
- Slant Height: Distance along the curved surface
Surface Area Formula
A = π(R² + r² + S(R + r))
Where:
- A = Total surface area
- R = Base radius (larger)
- r = Top radius (smaller)
- S = Slant height
Component Areas
Large Base Area: πR²
Small Base Area: πr²
Lateral Area: πS(R + r)
Relationship Between Height and Slant Height
From Height to Slant Height: S = √(H² + (R - r)²)
From Slant Height to Height: H = √(S² - (R - r)²)
Note: The slant height must always be greater than the difference in radii.
Real-World Applications
🏗️ Engineering
Hoppers, silos, and material handling equipment often use frustum shapes for efficient flow.
🏠 Architecture
Lamp shades, decorative elements, and structural components with tapered designs.
🔬 Manufacturing
Calculating material requirements for conical containers, funnels, and industrial components.