Cavalieri’s Principle
Table Of Contents
🎬 Math Angel Video: Cavalieri’s Principle Explained
An Example of Cavalieri’s Principle
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🛎️ Imagine you have 10 slices of bread.
- On the left, the slices are stacked neatly.
- On the right, they are slanted.
Even though the shapes look different, the number and size of the slices are the same, so the total volume is unchanged.
This is a simple example to show how volume can stay the same, even if the shape looks different.
What is Cavalieri’s Principle?
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🛎️ Definition of Cavalieri’s Principle:
Cavalieri’s principle states that if two solids have the same height and same cross-sectional areas, they have the same volume, even if they look different.
In other words, two solids have the same volume if:
- They have the same height, and
- At every level, their cross-sectional areas are the same.
🛎️ Applying Cavalieri’s Principle:
Cavalieri’s Principle can be very helpful. For example, both stacks have a 10 cm by 10 cm base and are 12 cm high.
The neat stack on the left has a volume of:
$$ \text{Volume} = (10 \text{ cm})^2 \times 12\text{ cm} = 1200 \text{ cm}^3$$
Here’s the clever part: by Cavalieri’s Principle, the slanted stack on the right must also have the same volume.
❇️ Exam Tip:
This principle gives us a shortcut, instead of working out the volume of a complicated or tilted solid directly, we can compare it to a simpler ones.
Using Cavalieri’s Principle in 3D
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Cavalieri’s Principle doesn’t just work with stacks of slices, it also works for any 3D solids.
🛎️ What Cavalieri’s Principle Tells Us:
If two solids have the same height and the same cross-sectional area at every level, then their volumes are equal, even if the solids look very different.
🛎️ Using Cavalieri’s Principle for Solids:
In the picture, the two shapes both have the same height. At each level, their cross-sections match in area.
So, even though one looks curved and the other looks slanted, they must have the same volume.
🛎️ Why Cavalieri’s Principle Is Useful:
Cavalieri’s Principle is a powerful tool in geometry. It helps us compare and calculate volumes of unusual 3D shapes.
🍪 Quiz: Practice Cavalieri’s Principle in 3D Shapes
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