Area of a Frustum of a Cone Calculator

Enter the top radius, bottom radius, and height of a conical frustum to get its slant height, lateral surface area, and total surface area (A = π(r+R)l + πr² + πR²).

Quick Facts

Slant height
l = √((R − r)² + h²)
Follows from the Pythagorean theorem using the radius difference and height.
Lateral surface area
A = π(r + R)l
The area of the sloped side only, excluding the top and bottom circles.
Total surface area
A = π(r + R)l + πr² + πR²
Lateral area plus the two flat circular end caps.

Your Results

Calculated
Lateral Surface Area
-
A = π(r + R)l
Total Surface Area
-
Lateral area + πr² + πR²
Slant Height
-
l = √((R − r)² + h²)
Top + Bottom Circle Area
-
πr² + πR²

Ready

Enter the top radius, bottom radius, and height, then press Calculate.

Formula and Method for the Area of a Frustum of a Cone

A frustum of a cone is the solid left when a cone is sliced by a plane parallel to its base and the pointed top is removed, leaving two parallel circular faces of different radii connected by a sloped side. This calculator finds the slant height, the lateral (side) surface area, and the total surface area from the top radius r, the bottom radius R, and the vertical height h.

How the calculation works

First, the calculator finds the slant height l — the straight-line distance along the sloped side between the top and bottom edges — using the Pythagorean theorem on the vertical height and the difference between the two radii: l = √((R − r)² + h²). The lateral surface area, which is the sloped band around the outside only, is then Alateral = π(r + R)l. Total surface area adds the two flat circular end caps: Atotal = π(r + R)l + πr² + πR². If the top radius is entered as 0, the shape becomes a full cone and the formulas simplify to the standard cone lateral area πRl and total area πRl + πR².

Common mistakes

  • Confusing slant height with vertical height: the slant height l is always longer than the vertical height h (unless the two radii are equal), because it runs diagonally along the side rather than straight up.
  • Radius vs. diameter: this formula uses radii, not diameters. If you measured across the top or bottom (diameter), divide by 2 before entering the value.
  • Mixing units: enter the top radius, bottom radius, and height in the same unit — results are in that unit squared (e.g., ft² if you entered feet).
  • Forgetting the end caps: lateral area alone (the sloped side) is smaller than total surface area, which also counts the top and bottom circles — use lateral area for material like sheet metal wrapped around the side, and total area when the ends are also covered.

Real-world applications

  • Sizing sheet metal, fabric, or insulation to wrap lampshades, buckets, funnels, and traffic cones (lateral surface area).
  • Estimating material for silos, hoppers, and tapered tanks, including the circular top and bottom (total surface area).
  • Architecture and product design for tapered columns, planters, and packaging.
  • Engineering surface-area-to-volume calculations for heat transfer or coating coverage on conical components.

Frequently Asked Questions

What is the formula for the lateral surface area of a frustum of a cone?
The lateral (side) surface area is A = π(r + R)l, where r is the top radius, R is the bottom radius, and l is the slant height. The slant height is l = √((R − r)² + h²), found from the height h using the Pythagorean theorem.
What is the total surface area formula for a frustum?
Total surface area adds the two circular end caps to the lateral area: A = π(r + R)l + πr² + πR², where r and R are the top and bottom radii and l is the slant height.
How do I find the slant height of a frustum?
The slant height l is the straight-line distance along the side between the top and bottom edges: l = √((R − r)² + h²), where R and r are the bottom and top radii and h is the vertical height. It is not the same as the vertical height unless R equals r.
What happens if the top radius equals the bottom radius or is zero?
If the top radius equals the bottom radius, the shape is a cylinder and the slant height equals the vertical height. If the top radius is zero, the frustum becomes a full cone and the lateral area formula reduces to A = πRl.

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