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## High school geometry

### Course: High school geometry > Unit 9

Lesson 2: Cavalieri's principle and dissection methods- Cavalieri's principle in 2D
- Cavalieri's principle in 3D
- Cavalieri's principle in 3D
- Apply Cavalieri's principle
- Volume of pyramids intuition
- Volume of a pyramid or cone
- Volumes of cones intuition
- Using related volumes
- Use related volumes
- Volume of prisms and pyramids

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# Volume of a pyramid or cone

CCSS.Math: ,

Where does the 1/3 come from in the formula for the volume of a pyramid? How does the volume of a cone relate? What about oblique pyramids (pyramids that lean to the side)?

## What are pyramids and cones?

A

**pyramid**is the collection of all points (inclusive) between a polygon-shaped base and an apex that is in a different plane from the base.Another way to think of a pyramid is as a collection of all of the dilations of the base with the apex as the center of dilation with scale factors from 0 to 1.

A

**cone**is a common pyramid-like figure where the base is a circle or other closed curve instead of a polygon. A cone has a curved lateral surface instead of several triangular faces, but in terms of volume, a cone and a pyramid are just alike.## Volume of a pyramid

The formula for the volume V of a pyramid is V, equals, start fraction, 1, divided by, 3, end fraction, left parenthesis, start text, b, a, s, e, space, a, r, e, a, end text, right parenthesis, left parenthesis, start text, h, e, i, g, h, t, end text, right parenthesis. Where does that formula come from?

### Where does the start fraction, 1, divided by, 3, end fraction come from in the formula?

Suppose we start with a cube with a side length of 1 unit. We can slice that cube into 3 congruent pyramids.

### Scaling the pyramid

Scaling a pyramid works exactly the same way as scaling the prism that encloses it does. When we scale a pyramid with volume V, start subscript, start text, u, n, s, c, a, l, e, d, end text, end subscript by factors of r, s, and t in three perpendicular directions, then the volume V, start subscript, start text, s, c, a, l, e, d, end text, end subscript of the scaled figure is V, start subscript, start text, u, n, s, c, a, l, e, d, end text, end subscript, r, s, t.

**Key idea:**The volume of pyramid is still start fraction, 1, divided by, 3, end fraction of the volume of the prism that encloses it, even after we scale them both.

### Sliding the slices

Imagine we sliced the pyramid into layers parallel to its base. We could slide those layers without changing the volume. As the number of layers gets close to infinity, our reshaped pyramid smooths out.

**Cavalieri's principle**says that as long as we don't change the height or the areas of the pyramid's cross-sections parallel to its base, we don't change the volume either! We can use the same formula for pyramid volume no matter where we move the apex.

### Changing the base shape

There's another really fascinating application of Cavalieri's principle to pyramids. Two bases can have the same area and entirely different shapes. If the height and base area of two pyramids or solids are equal, then so are their volumes, because the areas of all of the other cross-sections parallel to the base must be equal too.

So our formula V, start subscript, start text, p, y, r, a, m, i, d, end text, end subscript, equals, start fraction, 1, divided by, 3, end fraction, left parenthesis, start text, b, a, s, e, space, a, r, e, a, end text, right parenthesis, left parenthesis, start text, h, e, i, g, h, t, end text, right parenthesis works, no matter what 2D shape the base has.

### Getting start fraction, 1, divided by, 3, end fraction another way

Another way that mathematicians like you have convinced themselves that the volume of a pyramid is start fraction, 1, divided by, 3, end fraction the volume of the prism that encloses it is by approximating the volume using prisms.

We can model a pyramid as a stack of prisms, like building a pyramid out of blocks. This model has a volume that is a greater than the pyramid's volume. As we get more, thinner layers, we get closer and closer to the volume of the pyramid.

Number of layers | start fraction, start text, V, o, l, u, m, e, space, o, f, space, b, l, o, c, k, space, p, y, r, a, m, i, d, space, a, p, p, r, o, x, i, m, a, t, i, o, n, end text, divided by, start text, V, o, l, u, m, e, space, o, f, space, p, r, i, s, m, end text, end fraction |
---|---|

4 | approximately equals, 0, point, 469 |

16 | approximately equals, 0, point, 365 |

64 | approximately equals, 0, point, 341 |

256 | approximately equals, 0, point, 335 |

1024 | approximately equals, 0, point, 334 |

4096 | approximately equals, 0, point, 333 |

infinity | start fraction, 1, divided by, 3, end fraction |

Since prism-like figures can have any closed, 2D figure for the base, and since we can slide the prisms without changing their volume, the ratio holds true for all pyramid-like figures, including cones.

## Want to join the conversation?

- This is a great and helpful article, but 1/4 down the page, it says Vrst when I believe it should say V=rst since the volume=rst.(9 votes)
- problem 3: What is the volume of the pyramid?

why is the answer in cubic meters though

problem 4.2 What is the exact volume of the cone?

There is no pi in the input form so answering the question correctly is impossible for me(3 votes)- the m^2 is wrong, you should probably report that issue

to get pi, simply type "pi" and it should show up in the box(3 votes)

- So l cant use the formular of triangle(1 vote)
- A triangle is a 2-dimensional shape. We can do perimeter or area of a triangle, but not volume.

Volume is a measurement of space occupied by a 3-dimensional shape. So, different formulas are needed.(1 vote)

- How does the volume of a square pyramid

change if the base side length is increased

by a factor of 9 and the height is

unchanged?(1 vote) - Can you reverse the equation to find the volume of a cube?(0 votes)
- How i do this?(0 votes)
- :) Smile of likes(0 votes)
- How many more questions are there?(0 votes)
- There are only 6 questions on this sheet! :)(0 votes)