When you hear “pyramid,” your mind probably jumps to the iconic monuments of ancient Egypt—sharp summits, wide bases, and flawless geometric precision. But what happens if that top peak is removed? That’s where the fascinating world of truncated pyramids begins—and yes, understanding them involves some math.
For other 3D shapes like cones, cylinders, or spheres, head to the Volume Calculator for a full set of quick tools.
What is a Truncated Pyramid?
A truncated pyramid, also known as a frustum of a pyramid, is a pyramid with its top cut off by a plane that runs parallel to the base. The result is a solid with two parallel bases—one smaller, one larger—and trapezoidal faces connecting the sides.
This shape isn’t just a geometry concept; it appears in real-world design, from architecture and packaging to 3D modeling.
Take the glass pyramid at the Louvre Museum in Paris, for example. If you sliced off the top third, leaving a flat surface, you’d be looking at a classic truncated pyramid. Try out Circle Square Footage Calculator
💡 Did you know? The term “frustum” comes from Latin, meaning “piece” or “morsel.” It’s a common design in aerospace too—NASA’s Apollo command modules were shaped like frustums to help with safe reentry into Earth’s atmosphere.
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The Truncated Pyramid Volume Formula
Let’s get straight to the point: there’s a simple formula that makes finding the volume of a truncated pyramid easier than you might think. Here it is:
Volume = (1/3) × height × (area of bottom base + area of top base + √(bottom base area × top base area))
Or more compactly:
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 builds on the regular pyramid volume formula but adjusts for having two bases instead of one. It averages the two base areas, and the square root term accounts for the tapering shape, making it more precise than a basic average.
Let’s look at an example with a truncated square pyramid:
Now plug the values 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’ve just calculated the volume of a frustum. Whether you're designing a 3D object, estimating material needs, or solving a math problem, this formula is a handy tool to keep in your back pocket. Working with box-shaped structures? The Rectangular Prism Volume Calculator is great for calculating volume with length, width, and height.
Truncated Geometry in Real Life and Culture
Geometry isn’t limited to textbooks—it shows up all around us, from ancient ruins to video game worlds.
Take Minecraft, for instance. Players often build pyramid-shaped structures with flat tops for towers or platforms. Without even thinking about it, they’re creating truncated pyramids, applying real-world geometry in their digital builds.
Now rewind a few thousand years to ancient Mesopotamia. The Great Ziggurat of Ur is a perfect historical example. Unlike the pointed pyramids of Egypt, ziggurats featured flat tops and step-like layers. These platforms weren’t just aesthetic—they were functional too, often supporting temples or altars.
Whether it’s in virtual environments or ancient architecture, the truncated pyramid has been a go-to design for centuries. And today, knowing how to calculate its volume? Still just as practical.
Check out Math section to solve math quickly and easily
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