A conical frustum is what you get when a cone is sliced horizontally, creating two circular faces and one curved surface between them. These shapes are more common than they sound: lampshades, cups, funnels, ventilation systems, and even ancient architecture use frustums to combine strength with efficient volume. But calculating the surface area of a conical frustum isn’t as straightforward as it is for spheres or cubes. It requires a combination of circular and slant-based geometry. In this article, we’ll walk through what a conical frustum is, how its surface area is derived, how it compares with other shapes, and where it shows up in the real world — complete with formulas, use cases, and even a historical fun fact or two.
What Is a Conical Frustum?
Imagine slicing the top off a cone — what you’re left with is called a conical frustum. It’s a 3D shape with two circular bases (one larger, one smaller) and a slanted, curved surface connecting them. This isn’t just a theoretical figure you meet in geometry class — conical frustums are everywhere, from paper cups and traffic cones to lampshades, buckets, and industrial ducts.
The word frustum comes from Latin, meaning “a piece broken off.” And that’s exactly what it is — a cone with the top removed by a plane parallel to the base. The result? A solid that blends both flat and curved geometry, ideal for designs where full cones aren’t practical.
You might wonder: why don’t we just call it a “truncated cone”? In casual settings, we often do. But in mathematics, engineering, and architecture, “frustum” is the preferred term, not only for cones but also for pyramids. It reflects a more precise geometric distinction used in professional fields, especially when calculating surface area or volume, where slant height plays a crucial role.
So the next time you hold a cup or look at a cooling tower, you’re holding a frustum — a deceptively simple shape with a lot going on underneath.
How the Formula Works
The surface area of a conical frustum might look intimidating at first, but it’s surprisingly logical once you break it down. The shape has three distinct parts to consider: the curved lateral surface, and the two circular faces — one at the top and one at the bottom.
Here’s the formula
A=π(r1+r2)s+πr12+πr22
Where:
-
r1 is the radius of the bottom base -
r2 is the radius of the top base -
sis the slant height, or the length along the side -
The first term gives the lateral surface area, and the last two are the areas of the two circular bases
If this looks familiar, it’s because it builds on the surface area of a full cone — but with the top sliced off. One way to derive the frustum formula is by subtracting the surface area of the missing top cone from the larger original cone.
It’s a perfect example of geometry in action: a clean shape, sliced with precision, and explained through math.
If No Slant Height
You don’t always get the perfect set of numbers when solving geometry problems. Sometimes, the slant height of a conical frustum isn’t provided, but that doesn’t mean you're stuck. With a little help from the Pythagorean Theorem, you can calculate it yourself.
Picture a right triangle formed by the frustum’s vertical height and the difference in the radii of the two circular bases. The slant height is the hypotenuse. Here's the formula to find it:
s=(r1−r2)2+h2
Where:
-
r1andr2are the radii of the lower and upper circles -
his the vertical height (not slant height!) -
sis the slant height you’re solving for
Once you find sss, you can plug it back into the full surface area formula and proceed as usual.
Hadrian’s Hidden Geometry
When Roman Emperor Hadrian rebuilt the Pantheon around 118 AD, he wasn’t just reviving a temple—he was creating a geometric masterpiece. The building’s iconic dome isn’t a perfect hemisphere or a cone, but something in between: a form that closely resembles a conical frustum.
This design wasn’t just aesthetic. Scholars believe the dome’s frustum-like shape helped distribute weight more efficiently, reducing outward pressure on the massive walls.¹ Combined with ancient Roman concrete engineering, it allowed the dome to remain the largest unreinforced concrete dome in the world, even after nearly 2,000 years.
Though Hadrian may not have used modern formulas, his architects clearly understood geometry in structure. The elegant balance of volume, surface, and strength shows how timeless math principles shape lasting design.
