A truncated cone is the shape you get when you slice the top off a cone with a straight cut, leaving a flat circular surface on top and a larger circular base on the bottom. Think of a paper cup, a lampshade, or a bucket. In geometry, this shape is closely related to a frustum, and the two terms are often used interchangeably in everyday conversation, though there is a subtle technical difference.
How It Relates to a Frustum
A frustum is specifically the portion of a cone (or pyramid) that sits between two parallel planes cutting through the solid. A truncated cone is slightly more general: the cutting plane doesn’t have to be parallel to the base. In practice, though, most people searching for either term are picturing the same thing: a cone with its pointy tip sliced off by a flat, parallel cut, producing a smaller circle on top and a larger circle on the bottom. That parallel-cut version is both a truncated cone and a frustum. The word “frustum” comes from Latin and literally means “morsel,” as in a piece cut from a larger whole.
Key Measurements
A truncated cone is defined by three measurements:
- Bottom radius (R): the radius of the larger circular base
- Top radius (r): the radius of the smaller circular top
- Height (h): the perpendicular distance between the two flat faces
From these three, you can derive everything else, including slant height, volume, and surface area. The slant height (s) is the distance along the angled side from the edge of the top circle to the edge of the bottom circle. You can calculate it using the Pythagorean theorem: the slant height equals the square root of the height squared plus the difference of the two radii squared. In notation, s = √[h² + (R − r)²].
How to Calculate Volume
The easiest way to think about the volume of a truncated cone is as the difference between two complete cones. Imagine extending the slanted sides upward until they meet at a point, forming one large cone. The piece you sliced off the top is a smaller cone. The volume of the truncated cone is simply the big cone’s volume minus the small cone’s volume.
The formula for the volume of any cone is (1/3) × π × r² × h. For a truncated cone specifically, this simplifies to a single formula:
V = (π × h / 3) × (R² + R × r + r²)
So if your bucket has a bottom radius of 15 cm, a top radius of 10 cm, and a height of 30 cm, the volume works out to (π × 30 / 3) × (225 + 150 + 100) = 10π × 475 ≈ 14,923 cubic centimeters, or about 14.9 liters.
How to Calculate Surface Area
The total surface area has three parts: the large circular base, the small circular top, and the sloping side (called the lateral surface). The full formula combines them:
SA = π × [s × (R + r) + R² + r²]
Here, s is the slant height, R is the larger radius, and r is the smaller radius. The term π × s × (R + r) gives the lateral surface area alone, which is useful if you only need the side, for instance when calculating how much material to wrap around a lampshade. The remaining terms, π × R² and π × r², are just the areas of the two circular faces.
Everyday Objects With This Shape
Truncated cones are everywhere once you start noticing them. Paper and plastic cups taper outward from bottom to top so they’re easy to stack and comfortable to hold. Buckets, flowerpots, and mixing bowls use the same geometry for stability and nesting. Lampshades spread light outward by flaring from a narrow top ring to a wider bottom edge. A fez, the brimless felt hat, is a classic truncated cone. Traffic cones are technically truncated cones too, with a flat top and a wider base. Even drinking glasses with a slight taper qualify.
The shape is popular in manufacturing because it combines structural strength with easy stacking. Objects with straight, angled sides can nest inside each other without getting stuck the way cylinders do, since the taper leaves a small gap between stacked items.
Making a Truncated Cone From Flat Material
If you need to build a truncated cone out of sheet metal, paper, or fabric, you’ll work with a flat pattern called a “development” or “net.” The flat shape that wraps into a truncated cone is a section of a ring (an annular sector). It looks like a wide arc cut from a donut shape.
To create this pattern, you first figure out where the apex of the original full cone would have been by extending the slant lines until they meet. That distance from the apex to the bottom edge gives you the outer arc radius, and the distance from the apex to the top edge gives you the inner arc radius. The angle of the arc depends on the circumference of the base relative to the full circle at that radius. Once you cut out this ring-shaped wedge and roll it, the inner arc becomes the top circle and the outer arc becomes the bottom circle. You then add separate circles for the top and bottom faces if your design needs them closed.
This flat-pattern method is standard in sheet metal work, tailoring, and paper craft. Getting the arc radii and sweep angle right is the only tricky part, and several free online calculators will generate the pattern from your three measurements.