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High school geometry

Course: High school geometry > Unit 9

Lesson 2: Cavalieri's principle and dissection methods
Math
High school geometry
Solid geometry
Cavalieri's principle and dissection methods
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Cavalieri's principle in 3D

Google Classroom
If two 3D figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.

Cavalieri's principle in 3D

Key idea: If two 3D figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.

Why it works

Imagine we have a stack of coins (or books, playing cards, or anything with parallel planes). If we push on the top of the stack so that it slopes to the side, have we changed the volume? Of course not!
2 stacks of identical poker chips. Both stacks have the same number of chips. One stack resembles a right cylinder. The other stack has each chip set off-center from the one below it.
By HB (Own work) [CC BY-SA 4.0 (http://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons
We can cut a solid into many parallel layers, then slide them from side to side without changing the volume.
Two right cylinders. The first cylinder starts out the same as the second cylinder. Then, it gets sliced into more and more layers. The layers slide left and right. When the layers are very thin, the first cylinder resembles an oblique cylinder.
Try out the Cavalieri's principle cylinder simulation for yourself. Drag the sliders to change the number slices and how far the cylinder on the left is skewed. Try increasing the number of slices until the cylinder looks smooth.

Exploring more unusual shapes

We can use Cavalieri's principle for more than just prisms and cylinders. For example, we can slide the layers of a cone from side to side without changing the volume, too.
Animation showing horizontal cross-sections of a right cone and a second figure with congruent cross-sections to the cone, but with the cross-sections sliding to the side in a wavy and irregular pattern.
Try the Cavalieri's sculpture simulation for yourself. Drag your mouse over the cone on the right to sculpt it. Notice that no matter how you sculpt, the cross-sectional areas of both figures at any given height remain equal.
Both figures have a height of 21 and a base area of 64π.
Problem 1
What is the volume of cone on the left?
  • Your answer should be
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cubic units
What is the volume of sculpted cone on the right?
  • Your answer should be
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
cubic units

Cavalieri's principle with different shapes

One of the more useful features of Cavalieri's principle is that it works even when the cross-sections have different shapes, as long as they still have equal areas.
Problem 2.1
The following figures all have the same height and same base area.
Which of the following figures have the same volume?
Choose all answers that apply:

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  • leafers ultimate style avatar for user Elizabeth Singh
    Posted 3 years ago. Direct link to Elizabeth Singh's post “How does the last option ...”
    How does the last option on the 1st question, the one with the hexagon as a base, have the same volume as the other two?
    (16 votes)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      Posted 3 years ago. Direct link to David Severin's post “The volume for a cone and...”
      Good Answer
      The volume for a cone and pyramid are the same, V = 1/3 Bh where B is the area of the base. So even though the base is a different shape, as long as the areas and heights are the same, they will have the same volume. Just because the way to find the areas differ by the shape of the base, it does not change the area of the bases. Of course, it would be hard to get a circle and a pentagon to have the exact same area because this would mean the side of pentagon would have to be a multiple of π.
      (18 votes)
  • blobby green style avatar for user hayo2999
    Posted 3 years ago. Direct link to hayo2999's post “The last statement on thi...”
    The last statement on this article i believe is incorrect. Suppose you have a cylinder with a base area of 5 and a height of three and a cone of a base area of 5 and a height of 3 they will have a different volume. The principle should work with the same type of shapes and not completely different shapes as they all have different volume formulas.
    (0 votes)
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    • leaf red style avatar for user John Thompson
      Posted 3 years ago. Direct link to John Thompson's post “Cavalieri's wouldn't appl...”
      Good Answer
      Cavalieri's wouldn't apply to a cone and cylinder because while the base of each might be equal, you could not take cross-sections at an arbitrary height and show them to be equal. This is why the language of Cavalieri's Principle is specific to areas of cross-sections and not just areas of bases.

      Now as long as the areas of cross-sections are equal then the shape of those cross-sections is irrelevant. This is actually one of the ways in which you can derive the formula for the volume of a sphere... it is equivalent to a cylinder that has been hollowed out by cones (think inverse of an hourglass). It compares areas of circles to areas of rings.
      (13 votes)
  • aqualine seed style avatar for user Russell LVP
    Posted 3 months ago. Direct link to Russell LVP's post “I’m on mobile, so not man...”
    I’m on mobile, so not many options, how do I express pi.
    (4 votes)
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  • mr pants orange style avatar for user Remedy
    Posted 2 months ago. Direct link to Remedy's post “In the problem 2.1, I sti...”
    In the problem 2.1, I still cannot understand why these shapes can have the same volume. Is the volume of the pentagon-based pyramid the same as the other 2 cones even though they use pi?
    (1 vote)
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    • blobby green style avatar for user iwil
      Posted 24 days ago. Direct link to iwil's post “Yes, the same volume. Li...”
      Yes, the same volume.

      Like you said, to calculate the cones, we have to use pi, because:
      to calculate the base, which is a circle, area=pi*r^2

      But the problem said: “The following figures all have the same height and same base area.”.
      Let’s make the area of the circle equals pi, so the area of pentagon is also pi.
      As we know, volume = 1/3 * area * height, (or the Cavalieri’s principle)
      plus, they share the same area & height,
      so the volumes are the same.

      (English is not my first language and I hope it helps you.)
      (2 votes)