Area of a Frustum of a Cone Calculator

A frustum of a cone is created when you slice off the top portion of a full cone with a plane parallel to its base. The total surface area consists of three parts: the circular bottom base, the circular top face, and the lateral curved surface connecting them. Mathematically, the total surface area (A) is calculated using the formula: A = π × (r₁² + r₂² + s × (r₁ + r₂)), where r₁ is the base radius, r₂ is the top radius, and s is the slant height. If you only know the vertical height (h) instead of the slant height, the tool first computes s using the Pythagorean theorem: s = √(h² + (r₁ - r₂)² ).

Let us walk through a practical calculation. Suppose you have a truncated cone with a large base radius (r₁) of 8 units, a top radius (r₂) of 2 units, and a slant height (s) of 12 units. First, square the radii: 8 squared is 64, and 2 squared is 4. Next, add the two radii together: 8 + 2 = 10. Multiply this sum by the slant height: 10 × 12 = 120. Now, sum these three components together inside the formula: 64 (base) + 4 (top) + 120 (lateral) = 188. Finally, multiply by Pi (π ≈ 3.14159), which gives a total surface area of approximately 590.62 square units.

When measuring your physical objects or drafting designs, always double-check whether you have been given the vertical height or the slant height, as using the wrong one will skew your final area output. Maintain consistent units of measurement—mixing centimeters and inches will lead to erroneous results. Finally, remember that total surface area includes both the top and bottom circular faces; if your specific engineering application involves an open-topped container, you will need to subtract the area of the top circle (π × r₂²) manually.