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

🪣

Buckets

Tapered shape for easy stacking

💡

Lampshades

Conical shade for directed light

🏗

Cooling Towers

Hyperboloid frustum shape

🥤

Cups & Glasses

Tapered for comfortable grip

🌺

Flower Pots

Wider base for stability

🎥

View Frustum

3D camera clipping volume

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

What is a frustum?

A frustum is a portion of a solid, usually a cone, that lies between two parallel planes cutting it. When you slice the top off a cone with a plane parallel to the base, the remaining solid is called a frustum or truncated cone. It has two circular faces of different sizes connected by a sloped lateral surface.

How do you calculate the volume of a frustum?

The volume of a frustum is V = (π × h / 3)(R² + R × r + r²), where R is the bottom radius, r is the top radius, and h is the perpendicular height. This formula works for any frustum including the special cases where r = 0 (cone) or R = r (cylinder).

What is the lateral surface area of a frustum?

The lateral surface area of a frustum is A = π × (R + r) × l, where l is the slant height. The slant height is calculated as l = √(h² + (R − r)²). This gives the area of just the curved side surface, not including the top and bottom circular faces.

What is the difference between a frustum and a cone?

A cone tapers to a single point (apex), while a frustum is a cone with its top cut off by a plane parallel to the base, resulting in two circular faces of different sizes connected by a sloped lateral surface. A frustum with top radius r = 0 is a complete cone.

Where are frustums used in real life?

Frustums appear in many everyday objects: buckets, lampshades, drinking cups, flower pots, traffic cones, cooling towers, and architectural columns. In computer graphics, the view frustum defines the visible 3D space of a camera. In civil engineering, frustum shapes are used in foundation design and dam construction.

Reference this content, page, or tool as:

"Frustum Calculator" at https://MiniWebtool.com/frustum-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-02

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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