I'm confused: Is Cavalieri's principle related to density? And if so, how?

What *alex81* said. _(in comments)_ Cavalieri's principle has no mention of density and only works with Volume, Area, and Lengths of the 3D shapes. No Quantities.

wait, I'm confused, what is the difference between area and volume density and how do i know which formula to use?

Area is simply volume in 2D. To figure out the density for a 3D object, let's say a Rectangular-Based Pyramid, we need to find the volume first. The formula is 1/3*(l*w)*h. The (l*w) is the area of the base of the Pyramid. To find the volume you would multiply the base area by the height, and then by 1/3. Furthermore, to find the density you would need the quantity and the volume you just solved for to find the density in this equation: Quantity / Volume = Density To answer your question, you can tell which formula to use by whether it's 3D or 2D. Density is just the next step after finding the area. (You can also find the density in a 2D object: Quantity / Area = Density) Hope this helps!

Why does density work for 2D and 3D objects the same?

The FAQ doesn't mention the density of an object with an area, but it should be the same except for (Volume unit)^3, it would be (Volume unit)^2. Theoretically though.

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Course: High school geometry > Unit 9

Lesson 4: Density

Solid geometry FAQ

Google Classroom

Frequently asked questions about solid geometry

Why do we need to know the difference between 2D and 3D objects?

Understanding the difference between 2D and 3D objects is important for solid geometry. 2D shapes, like squares and circles, only have length and width. 3D shapes, like cubes and spheres, also have depth. In order to find the volume and surface area of 3D shapes, we need to be able to identify them.

What is Cavalieri's principle?

Cavalieri's principle states that two solids with the same height and cross-sectional area at every height have the same volume. This principle is useful for determining the volume of more complex shapes by breaking them down into simpler shapes that we already know how to find the volume of.

How do we find the volume of a prism?

To find the volume of a prism, we use the formula

V = B h

, where:

$V$ ‍ is the volume
$B$ ‍ is the area of the base of the prism
$h$ ‍ is the height of the prism

How do we find the volume of a pyramid?

To find the volume of a pyramid, we use the formula

V = \frac{1}{3} B h

, where:

$V$ ‍ is the volume
$B$ ‍ is the area of the base of the pyramid
$h$ ‍ is the height of the pyramid

How do we find the surface area of a prism or pyramid?

To find the surface area of a prism (or a pyramid), we add up the areas of all of its faces.

What is density and why is it important?

Density is a measure of how compact a material is. It is defined as the mass of the material divided by its volume. In solid geometry, we can use density to compare different materials and determine which one is heavier or lighter for a given volume.

How are these topics used in the real world?

There are countless applications of solid geometry in the real world! For example, architects and engineers use concepts like volume and surface area to design buildings, bridges, and other structures. In manufacturing, density is an important consideration when selecting materials. And in science, researchers often use dissection methods to study the internal structure of an object.

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SuperSwetter
Posted a year ago. Direct link to SuperSwetter's post “I'm confused: Is Cavalier...”
I'm confused: Is Cavalieri's principle related to density? And if so, how?
Button navigates to signup pageComment on SuperSwetter's post “I'm confused: Is Cavalier...”
(10 votes)
Answer
- ⓉⓤⓡⓣⓛⓔⓂ①②
  Posted a year ago. Direct link to ⓉⓤⓡⓣⓛⓔⓂ①②'s post “What *alex81* said. _(in ...”
  What alex81 said. (in comments)
  
  Cavalieri's principle has no mention of density and only works with Volume, Area, and Lengths of the 3D shapes. No Quantities.
  Comment on ⓉⓤⓡⓣⓛⓔⓂ①②'s post “What *alex81* said. _(in ...”
  (7 votes)
Umberon2000
Posted 8 months ago. Direct link to Umberon2000's post “wait, I'm confused, what ...”
wait, I'm confused, what is the difference between area and volume density and how do i know which formula to use?
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(5 votes)
Answer
- LemonTurtle
  Posted 7 months ago. Direct link to LemonTurtle's post “Area is simply volume in ...”
  Area is simply volume in 2D. To figure out the density for a 3D object, let's say a Rectangular-Based Pyramid, we need to find the volume first.
  
  The formula is 1/3*(l*w)*h. The (l*w) is the area of the base of the Pyramid. To find the volume you would multiply the base area by the height, and then by 1/3. Furthermore, to find the density you would need the quantity and the volume you just solved for to find the density in this equation:
  
  Quantity / Volume = Density
  
  To answer your question, you can tell which formula to use by whether it's 3D or 2D. Density is just the next step after finding the area. (You can also find the density in a 2D object: Quantity / Area = Density)
  
  Hope this helps!
  Button navigates to signup page
  (6 votes)
28sdrown
Posted a year ago. Direct link to 28sdrown's post “Why does density work for...”
Why does density work for 2D and 3D objects the same?
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(3 votes)
Answer
- Ian Milligan
  Posted 7 months ago. Direct link to Ian Milligan's post “The FAQ doesn't mention t...”
  The FAQ doesn't mention the density of an object with an area, but it should be the same except for
  (Volume unit)^3, it would be (Volume unit)^2. Theoretically though.
  Button navigates to signup page
  (2 votes)
BatzlerA
Posted a year ago. Direct link to BatzlerA's post “Im confused on density”
Im confused on density
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(4 votes)
Answer
NFinstein
Posted a year ago. Direct link to NFinstein's post “typo: How do we find the ...”
typo: How do we find the volume of a prism?
To find the volume of a pyramid, we use the prism V=Bh
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(2 votes)
Answer
- Josh Meyers
  Posted a month ago. Direct link to Josh Meyers's post “A pyramid is V=1/3BH, a p...”
  A pyramid is V=1/3BH, a prism is V=BH
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  (1 vote)
Sam Harris
Posted a year ago. Direct link to Sam Harris's post “Cavaliers prinicple and d...”
Cavaliers prinicple and density do not make sense t me.
Button navigates to signup pageComment on Sam Harris's post “Cavaliers prinicple and d...”
(1 vote)
Answer
- samandjam
  Posted a month ago. Direct link to samandjam's post “me neither man, I just kn...”
  me neither man, I just know what to do, I don't know why.
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  (1 vote)
sabrina.mamedova
Posted a month ago. Direct link to sabrina.mamedova's post “what's the difference bet...”
what's the difference between area density and volume density and how do you find each?
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(1 vote)
Answer
- Lincoln Earley
  Posted 4 days ago. Direct link to Lincoln Earley's post “Area Density is used for ...”
  Area Density is used for 2 dimensional objects like a rectangle. If the area of the rectangle is, say, 1.2m^2 (square meters) and the quantity of something (x) is, say, 2x then the Area density is going to be about 1.7(x/m^2), (1.7 of that something for every square meter of area.)
  
  Volume Density is the 3D version of Area Density. You can imagine volume as 3D area. You just multiply the 1.2m^2 (square meters) by the height, say it's 2m, and you get a volume of 2.4m^3 (cubic meters), and you would just do the same thing to find the density of the rectangular prism and get D=0.8(x/m^3), (0.8 of that something for every square meter of area.)
  
  Area is 2-dimensional volume.
  Volume is 3-dimensional area.
  But at the end of the day, they're the same concept.
  Hope this helps!
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  (1 vote)
Pamplemousse010
Posted 2 months ago. Direct link to Pamplemousse010's post “What is the relationship ...”
What is the relationship between density and volume?
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(1 vote)
Answer
- szhang045
  Posted 2 months ago. Direct link to szhang045's post “density = mass/ volume”
  density = mass/ volume
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  (1 vote)