Frustum Calculator
Calculate the volume, lateral surface area, total surface area, and slant height of a frustum (truncated cone). Enter the top radius, bottom radius, and height to get instant results with step-by-step formulas and an interactive 3D diagram.
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About Frustum Calculator
The Frustum Calculator computes the volume, lateral surface area, total surface area, slant height, and centroid of a frustum (truncated cone). A frustum is formed when a cone is cut by a plane parallel to its base, producing a solid with two circular faces of different sizes. Enter the bottom radius (R), top radius (r), and height (h) to get instant results with step-by-step formulas and an interactive cross-section diagram.
Real-World Applications of Frustums
Key Formulas for a Frustum
For a frustum with bottom radius R, top radius r, and perpendicular height h:
| Property | Formula | Description |
|---|---|---|
| Slant Height | \(l = \sqrt{h^2 + (R - r)^2}\) | Length along the lateral side |
| Volume | \(V = \frac{\pi h}{3}(R^2 + Rr + r^2)\) | Enclosed 3D space |
| Lateral Area | \(A_l = \pi(R + r) \times l\) | Curved side surface only |
| Top Base Area | \(A_{top} = \pi r^2\) | Smaller circular face |
| Bottom Base Area | \(A_{bot} = \pi R^2\) | Larger circular face |
| Total Surface Area | \(A_t = A_l + \pi R^2 + \pi r^2\) | All surfaces combined |
Understanding Frustum Geometry
A frustum is created by slicing a right circular cone with a plane parallel to its base. The height h is the perpendicular distance between the two parallel circular faces. The slant height (l) is the distance along the lateral surface between the edges of the two bases — it is longer than h because it runs diagonally. When the top radius equals zero (r = 0), the frustum becomes a complete cone. When both radii are equal (R = r), the frustum becomes a cylinder.
How to Use the Frustum Calculator
- Enter the bottom radius (R): Type the radius of the larger circular base, or click a quick example like Bucket, Lampshade, or Cup.
- Enter the top radius (r): Type the radius of the smaller circular top. Set to 0 for a complete cone.
- Enter the height (h): Type the perpendicular height of the frustum.
- Click Calculate Frustum: Press the button to compute all properties instantly.
- Review the results: See volume, surface areas, slant height, and centroid in the results cards. Toggle the diagram overlays to visualize dimensions, slant height, the original cone extension, and centroid location.
Frustum vs. Cone vs. Cylinder
A cone has one circular base and tapers to a point (apex). A cylinder has two equal circular bases connected by a straight lateral surface. A frustum sits between these two shapes: it has two unequal circular bases connected by a sloped surface. The frustum volume formula \(V = \frac{\pi h}{3}(R^2 + Rr + r^2)\) generalizes to the cone formula when r = 0 and to the cylinder formula when R = r.
Centroid of a Frustum
The centroid (center of mass for uniform density) of a frustum is located at height \(\bar{y} = \frac{h(R^2 + 2Rr + 3r^2)}{4(R^2 + Rr + r^2)}\) measured from the larger base. This is always closer to the larger base than to the smaller one, since more mass is concentrated near the wider end.
FAQ
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"Frustum Calculator" at https://MiniWebtool.com/frustum-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-02
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