Truncated Cone Calculator

A truncated cone, formally known as a frustum of a cone, is created when a plane slices through a cone parallel to its base, removing the upper apex portion. Calculating its properties relies on a few fundamental geometric formulas. The volume (V) depends on the height (h), bottom radius (r₁), and top radius (r₂), expressed as V = (1/3) π h (r₁² + r₂r₁ + r₂²). To find the slant height (s) used in surface area calculations, the Pythagorean theorem is applied to the vertical height and the difference between the two radii: s = √((r₁ - r₂)² + h²). Once the slant height is established, the lateral surface area is computed via A_lat = π (r₁ + r₂)s, and adding the areas of both circular bases yields the total surface area.

Let us walk through a practical example to see how the equations operate in real life. Imagine you are building a custom industrial funnel shaped like a truncated cone with a bottom radius of 5 cm, a top radius of 2 cm, and a vertical height of 4 cm. First, we calculate the slant height (s) using the formula s = √((r₁ - r₂)² + h²). Substituting our values gives s = √((5 - 2)² + 4²) = √(3² + 4²) = √(9 + 16) = √25 = 5 cm. Next, we determine the lateral surface area using A_lat = π (r₁ + r₂)s, which equals π (5 + 2)5 = 35π cm² (approx. 109.96 cm²). Finally, to find the volume, we apply the volume formula: V = (1/3) π (4) (5² + (2)(5) + 2²) = (4/3) π (25 + 10 + 4) = (4/3) π (39) = 52π cm³ (approx. 163.36 cm³).

When working with truncated cones, always ensure that your measurements use consistent units before plugging them into any formulas; mixing centimeters and meters will drastically skew your results. Pay close attention to whether you are given the radii or the diameters of the circular ends, as forgetting to divide the diameter by two is a frequent source of calculation errors. Finally, remember that the slant height is always longer than or equal to the vertical height because it travels diagonally along the outer slope of the frustum.