When most people hear the word pyramid, they picture the towering structures of ancient Egypt—sharp peaks, wide bases, and pure geometry in stone. But what happens when that peak gets chopped off? That’s when we enter the world of truncated pyramids—and yes, there’s math involved.
What is Truncated Pyramid?
A truncated pyramid (or more formally, a frustum of a pyramid) is exactly what it sounds like: a pyramid with the top sliced off by a plane parallel to its base. The result is a solid with two parallel bases—one smaller, one larger—and trapezoidal faces connecting them. It’s not just a theoretical curiosity; this shape shows up in everything from modern architecture to packaging design and 3D modeling.
Take, for instance, the Louvre Museum in Paris. Its iconic glass pyramid entrance looks like a full pyramid, but if it were designed with a flat top—imagine slicing off the top third—it would be a textbook truncated pyramid.
💡 Did you know? The shape of a truncated pyramid is also called a frustum. The word “frustum” comes from Latin, meaning “piece” or “morsel.” It's used in spacecraft design too—NASA’s Apollo command modules had frustum-like designs.
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The Truncated Pyramid Volume Formula (Made Simple)
Let’s cut to the chase: there’s a clean formula that makes calculating the volume of a truncated pyramid easier than it looks. Here it is:
Volume = (1/3) × height × (Area of bottom base + Area of top base + square root of (bottom base area × top base area))
Or in short:
V = (1/3) × h × (A₁ + A₂ + √(A₁ × A₂))
Where:
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V is the volume
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h is the vertical height between the two bases
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A₁ is the area of the bottom base
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A₂ is the area of the top base
This formula is a smart twist on the standard pyramid volume formula. Instead of just one base, you’re averaging two: the bottom and the top. The square root part (√A₁ × A₂) helps capture how the shape narrows, which makes the formula much more accurate than just averaging the two areas.
Let’s say you have a truncated square pyramid with:
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Bottom base: 10 x 10 inches → Area = 100 in²
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Top base: 4 x 4 inches → Area = 16 in²
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Height: 12 inches
Plug those numbers into the formula:
Volume = (1/3) × 12 × (100 + 16 + √(100 × 16))= (1/3) × 12 × (116 + √1600)= (1/3) × 12 × (116 + 40)= (1/3) × 12 × 156= 4 × 156= 624 cubic inches
And that’s it—you just calculated the volume of a frustum-shaped structure. Whether you’re working on a 3D model, estimating materials, or solving a geometry problem, this formula is your go-to.
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Truncated Shapes in Pop Culture and History
Geometry isn’t just for classrooms—it’s everywhere, including in video games and ancient architecture.
Take Minecraft, for example. Players often build pyramid-like structures but flatten the tops to make towers or platforms. That creates a truncated pyramid—without even realizing it, they’re using real-world geometry in their virtual worlds.
Now step back a few thousand years to ancient Mesopotamia. The Great Ziggurat of Ur is a famous example of this shape in history. Unlike the pointed Egyptian pyramids, ziggurats had flat tops, creating a step-like design. They were practical too—those flat tops held temples or altars.
From digital landscapes to ancient temples, truncated pyramids have been part of design for centuries. And knowing how to calculate their volume? Still just as useful.
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