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### 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

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 $V={\displaystyle \frac{1}{3}}(\text{base area})(\text{height})$ . Where does that formula come from?

### Where does the $\frac{1}{3}$ 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 ${V}_{\text{unscaled}}$ by factors of $r$ , $s$ , and $t$ in three perpendicular directions, then the volume ${V}_{\text{scaled}}$ of the scaled figure is ${V}_{\text{unscaled}}rst$ .

**Key idea:**The volume of pyramid is still

### 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 ${V}_{\text{pyramid}}={\displaystyle \frac{1}{3}}(\text{base area})(\text{height})$ works, no matter what 2D shape the base has.

### Getting $\frac{1}{3}$ another way

Another way that mathematicians like you have convinced themselves that the volume of a pyramid is $\frac{1}{3}$ 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 | |
---|---|

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.(14 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(4 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(2 votes)
- 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.(4 votes)

- For the question right below problem one "can we fit congruent pyramids to make a cube?" there was an animation that showed 3 congruent pyramids and the answered stated "Sadly, no. Here are 3 congruent pyramids that cannot fit together into a prism of any sort." My question is why didn't they choose 4 congruent pyramids and why can't 4 congruent pyramids make a cube?(2 votes)
- Think about it. If there were four pyramids, the fourth one would still not make a cube. There would still be 2 extra pyramids on the side missing. After watching the video, the reason we use three is because there are 3 types of pyramids that form the prism, one that divides z by 2, then one that divides x by two, and then finally one that divides y by two, because the height changes when forming each pyramid. If this is confusing, I strongly recommend watching the "Volume of pyramids intuition" video. Hope this helps.(2 votes)

- If the Oblate Spheroid is greater than or equal to 22/7 (PI) then where does quantum mechanics come into play?(2 votes)
- 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)- (b+9)^2 -> b^2 + 18b + 81

since the height stays, height*(18b + 81) is the additional resulting volume.(1 vote)

- Can you reverse the equation to find the volume of a cube?(1 vote)
- im in 6 grade(0 votes)
- Then you're doing advanced work! Good for you!(3 votes)

- How i do this?(0 votes)
- Just multiply the length of the base by the width of the base; multiply the answer by the height; divide the answer by three.(0 votes)

- :) Smile of likes(0 votes)