Cavalieri's Principle
Cavalieri's Principle says two solids have equal volume if they share the same height and cross-sectional area at every level. Slanting a stack of slices keeps its volume. So you can find each volume with V = base area × height.
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Cavalieri’s Principle in Real Life
- Imagine two stacks made from the same number of slices.
- If each slice has the same area, the stacks have the same volume.
- Rearranging the slices does not change the volume of the stack.
What Is Cavalieri’s Principle?
- Two solids can look different but still have the same volume.
- They must have the same height.
- They must have the same cross-sectional area at every level.
Use Cavalieri’s Principle to Find Volume
- At each height, a tilted prism has the same cross-sectional area as an upright prism with the same base area.
- So by Cavalieri’s principle they have the same volume, and you can use V = base area × height.
Checking Equal Volume Using Cavalieri’s Principle
- Compare the height and the cross-sectional area at matching levels.
- If they match all the way up, the solids have equal volume.
Practice Questions
Test your understanding
If two solids have the same height and identical cross-sectional areas at the same height, what can we conclude about their volumes?
Correct! 🎉 +10 pointsNot quite right
According to Cavalieri's principle, if two solids have the same height and their cross-sectional areas are identical at every height, they will have the same volume.
A cylinder has a height of and a cross-sectional area of . A rectangular prism has the same height of and a cross-sectional area of . Do they have the same volume?
Correct! 🎉 +10 pointsNot quite right
According to Cavalieri's principle, since both solids have the same height and cross-sectional area, they will have the same volume, regardless of the shape of the cross-sections.
Two solids have the same cross-sectional area of at every height. The first solid has a height of , and the second solid has a height of . Do they have the same volume?
Correct! 🎉 +20 pointsNot quite right
According to Cavalieri's principle, if two solids have the same height and their cross-sectional areas are identical at every height, they will have the same volume.
Two solids have a height of . One has a square cross-section with an area of , and the other has a circular cross-section with the same area of . What can you conclude about their volumes?
Correct! 🎉 +20 pointsNot quite right
According to Cavalieri's principle, the volumes of two solids will be the same if they have the same height and identical cross-sectional areas at every height, regardless of the shape of the cross-sections.
Two solids have the same height of . The first solid has a cross-sectional area of at every height, and the second solid has a cross-sectional area of at every height. Do they have the same volume?
Correct! 🎉 +20 pointsNot quite right
Although the solids have the same height, the cross-sectional areas differ. According to Cavalieri's principle, this means they will have different volumes.
A stack of toast has a height of , and its cross-sectional area is at each level. Another stack has the same height but a cross-sectional area of at each level. Using Cavalieri's principle, what can we conclude about the volumes of these two stacks?
Correct! 🎉 +30 pointsNot quite right
Since the heights are the same and the cross-sectional area of the second stack is larger, it will have a greater volume, according to Cavalieri's principle.
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Interactive Activity
Tilt a prism to see how Cavalieri's Principle keeps the volume the same regardless of slant
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Students Also Ask
The questions students bump into most on this topic
The two conditions are that both solids must have the same height, and that their cross-sectional areas parallel to the bases must be identical at every level. When both of these conditions hold, Cavalieri's principle guarantees that the two solids share the same volume.
It works because the volume of a solid is built up from all of its cross-sectional layers. If two solids share the same height and have identical cross-sectional areas at every level, they are made up of the same total amount of layers, so they must contain exactly the same volume.
You can apply Cavalieri's principle whenever you need to find the volume of an awkward or inclined solid. As long as you can match it to a simpler upright solid with the same height and the same cross-sectional areas at every level, the two solids will share the same volume.
Yes. As long as two solids share the same height and have identical cross-sectional areas parallel to the bases at every level, Cavalieri's principle guarantees that they have the same volume. The solids can look very different, yet still contain exactly the same amount of space.