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

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

So l cant use the formular of triangle

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.

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Course: Geometry (FL B.E.S.T.) > Unit 9

Lesson 5: Cavalieri's principle and dissection methods

Volume of a pyramid or cone

Google Classroom

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

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 = \frac{1}{3} (base area) (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.

Problem 1

What is the volume of each pyramid?

cubic units

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_{unscaled}

by factors of

r

s

, and

t

in three perpendicular directions, then the volume

V_{scaled}

of the scaled figure is

V_{unscaled} r s t

Let's consider what happens when we scale in one direction at a time.

When we scale the height of a prism, we're changing the number of layers. That makes sense, because each layer has the same cross-sectional area.

It looks a little odder to stack layers of a pyramid.

However, as we get thinner and thinner layers (and as we get closer to an infinite number of layers), the scaled pyramid starts to look exactly like we would expect. Since we've scaled the number of layers, the volume changes by the same scale factor.

We could repeat this process in each dimension.

Problem 2

The following pyramid is a scaled version of the previous square-based pyramid, using different scale factors for each dimension.

What is the volume of the rectangle-based pyramid?

{cm}^{3}

Key idea: The volume of pyramid is still

\frac{1}{3}

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.

Problem 3

What is the volume of the pyramid?

m^{3}

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.

Let's suppose both pyramid-like solids had a height of

h

and a base area of

B

We can think of the cross-sections of a pyramid-like solid as a collection of dilations of the base about the apex for all scale factors from

0

1

. Let's find the area of the cross-section after a dilation by scale factor

k

The dilation changes the lengths (like height) by

k

because they've scaled in

1

dimension and changes areas by

k^{2}

because they've scaled in

2

dimensions.

For any distance

k h

from the apex, the cross-sections of both figures have an area of

k^{2} \cdot B

. So Cavalieri's principle applies and the solids have equal volumes.

So our formula

V_{pyramid} = \frac{1}{3} (base area) (height)

works, no matter what 2D shape the base has.

Problem 4.1

The following pyramid has an isosceles right triangle as a base.

What is the volume of the pyramid?

{cm}^{3}

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	$\frac{Volume of block pyramid approximation}{Volume of prism}$ ‍
$4$ ‍	$\approx 0.469$ ‍
$16$ ‍	$\approx 0.365$ ‍
$64$ ‍	$\approx 0.341$ ‍
$256$ ‍	$\approx 0.335$ ‍
$1024$ ‍	$\approx 0.334$ ‍
$4096$ ‍	$\approx 0.333$ ‍
$\infty$ ‍	$\frac{1}{3}$ ‍

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.

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Genius71752
Posted 4 years ago. Direct link to Genius71752's post “This is a great and helpf...”
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.
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(14 votes)
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A A
Posted 3 years ago. Direct link to A A's post “problem 3: What is the vo...”
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
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(4 votes)
Answer
- Ash_001
  Posted 3 years ago. Direct link to Ash_001's post “the m^2 is wrong, you sho...”
  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
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  (3 votes)
rowenachari26
Posted a year ago. Direct link to rowenachari26's post “So l cant use the formula...”
So l cant use the formular of triangle
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(2 votes)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “A triangle is a 2-dimensi...”
  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.
  Comment on Kim Seidel's post “A triangle is a 2-dimensi...”
  (4 votes)
Braxton Fuller
Posted 3 months ago. Direct link to Braxton Fuller's post “If the Oblate Spheroid is...”
If the Oblate Spheroid is greater than or equal to 22/7 (PI) then where does quantum mechanics come into play?
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joyugo1967
Posted a year ago. Direct link to joyugo1967's post “How does the volume of a ...”
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?
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(1 vote)
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- dashpointdash
  Posted a year ago. Direct link to dashpointdash's post “(b+9)^2 -> b^2 + 18b + 81...”
  (b+9)^2 -> b^2 + 18b + 81
  since the height stays, height*(18b + 81) is the additional resulting volume.
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  (1 vote)
Munch
Posted 11 days ago. Direct link to Munch's post “For the question right be...”
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?
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(1 vote)
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- talinkumar
  Posted 15 hours ago. Direct link to talinkumar's post “Think about it. If there ...”
  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.
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Jorge Pimienta
Posted a year ago. Direct link to Jorge Pimienta's post “Can you reverse the equat...”
Can you reverse the equation to find the volume of a cube?
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romey22535
Posted 4 months ago. Direct link to romey22535's post “im in 6 grade”
im in 6 grade
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- Isabelle Sanchez
  Posted 4 months ago. Direct link to Isabelle Sanchez's post “Then you're doing advance...”
  Then you're doing advanced work! Good for you!
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Miranda,Felipe
Posted a year ago. Direct link to Miranda,Felipe's post “How i do this?”
How i do this?
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- Meryke Varmheldt
  Posted a year ago. Direct link to Meryke Varmheldt's post “Just multiply the length ...”
  Just multiply the length of the base by the width of the base; multiply the answer by the height; divide the answer by three.
  Comment on Meryke Varmheldt's post “Just multiply the length ...”
  (0 votes)
AustinR
Posted a year ago. Direct link to AustinR's post “:) Smile of likes”
:) Smile of likes
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- 🅗🅐🅝🅝🅐🅗 😜
  Posted 8 months ago. Direct link to 🅗🅐🅝🅝🅐🅗 😜's post “(^-^) Yeah!”
  (^-^)
  Yeah!
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  (3 votes)

Geometry (FL B.E.S.T.)