Frustum Surface Area Calculator – Truncated Cone
A frustum is the three-dimensional solid that remains after slicing the top off a cone with a cut parallel to the base. The result is a solid bounded by two circular faces of different sizes — a larger bottom base of radius R and a smaller top base of radius r — joined by a curved slanted surface. Buckets, drinking cups, grain hoppers, lamp shades, and conical roofs are all frustum shapes encountered in everyday life.
This calculator computes the lateral surface area, the top and bottom base areas, and the total surface area of a circular frustum. It also derives the slant height from the perpendicular height and the two radii, and converts the total surface area into seven common area units.
Key Formulas
Given a frustum with bottom radius R, top radius r, and perpendicular height h:
- Slant height:
l = √(h² + (R − r)²)— the length measured along the sloped side from the edge of the top base to the edge of the bottom base. - Lateral Surface Area (LSA):
LSA = π(R + r) × l— the area of the curved slanted surface only, excluding both circular caps. - Top Base Area:
A_top = π × r² - Bottom Base Area:
A_bottom = π × R² - Total Surface Area (TSA):
TSA = π(R + r)l + πR² + πr²— the complete enclosing area, lateral surface plus both caps.
How to Use This Calculator
Enter the bottom radius (R), top radius (r), and height (h) of your frustum, then select a length unit from the dropdown. All five output values update instantly as you type. Use the decimal places selector to control the precision shown, and click any copy icon to transfer an individual value to the clipboard, or use Copy All to export the full result set.
Practical Applications
Frustum surface area calculations arise across many fields:
- Manufacturing and fabrication: Estimating the sheet metal or fabric needed to produce funnels, hoppers, or conical containers requires accurate lateral surface area figures. Add the base circles when the top or bottom needs to be closed.
- Architecture: Conical tower roofs and decorative spires are frustum-shaped when the apex is removed. Surface area determines the cladding material required.
- 3D printing and CAD: Slicing software uses surface area to estimate material consumption and print time for hollow frustum-shaped objects.
- Packaging design: Tapered cardboard tubes and nested stackable cups are frustum shapes; surface area governs the amount of cardboard per unit.
- Heat transfer: Convective and radiative heat flux from a frustum-shaped body depends on the exposed surface area, especially the lateral area when the base is insulated.
Degenerate Cases
When the top radius is set to 0, the frustum becomes a complete cone. The calculator detects this and labels the result accordingly; the top base area becomes zero and the lateral surface area simplifies to πRl. When the top radius equals the bottom radius, the shape is a cylinder; the slant height equals the perpendicular height and the lateral area simplifies to 2πRh. Both cases are handled gracefully without errors.
Lateral vs. Total Surface Area: When to Use Each
Use the lateral surface area when only the sloping side matters — for example, when calculating the material for a conical roof that sits on top of a cylindrical wall (both bases are already accounted for elsewhere). Use the total surface area when you need to enclose the frustum completely, such as when manufacturing a sealed container or computing the total painted area of a frustum-shaped object.
Unit Conversion
All inputs are interpreted in the selected length unit and the resulting areas are expressed in the corresponding area unit (e.g., cm → cm²). The cross-unit conversion table beneath the main results shows the total surface area in mm², cm², m², km², in², ft², and yd² simultaneously. No intermediate rounding is applied during unit conversion, preserving the full floating-point precision of each result.
Validation and Constraints
The bottom radius must be strictly positive. The top radius must be non-negative and must not exceed the bottom radius — if it does, the calculator displays an error prompting you to swap the values. The height must be strictly positive. All values are constrained to a maximum of 1 × 10⁹ to avoid floating-point overflow.