Why is the Volume of sphere is divided by 3?

This is a really good video that I found: https://www.youtube.com/watch?v=xuPl_8o_j7k It explains the formula with simple geometry and algebra, and in animation.

who found the proof to the volume of sphere? better question, how?

Click on the line of text under the section "Volume of a Sphere" that reads: "Where does this formula come from?" It gives a, in my opinion, somewhat ambiguous answer, but it is an answer nonetheless.

When using the pi symbol in calculations, the answer comes out differently than if you just use 3.14, which is how the answers on here are calculated. Why not just recommended to use only 3.14 for pi, not the symbol on calculators as shown on the videos? It is frustrating to get sent back to zero when you use your calculator correctly but you should just use one standard calculation of 3.14.

3.14 is a simplified version of the real value of pi: 3.141592653589793238462643383279502884197169399375105820974944592307816406286 http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html

how can I find volume of a tetrahedron

A tetrahedron is just a triangular pyramid. If any pyramid has height h and a base of area b, then it's volume is bh/3.

01101001 00100000 01101100 01101111 01110110 01100101 00100000 01101101 01100001 01111000 00100000 01110000 01100101 01110100 01100101 01110010

01010111 01101000 01101111 00100111 01110011 00100000 01110100 01101000 01100001 01110100 00111111

What would be the difference between a triangular prism and a pyramid?

A prism has the base that goes through the whole figure, so a triangular prism will have two congruent bases that are a triangle. A triangular pyramid has a triangular base and each base vertex that has an edge that goes to a common vertex.

When we need to use the value of pi to find the area of a circle for part of the problem, which number value of pi would be better to use to find a more exact calculation? Would it be 3.14, the symbol for pi, or 22/7?

pi = 3.14156... 3.14 22/7 = 3.1428... So 3.14 will give a slightly underestimate it, and 22/7 will slightly overestimate it. The 15 in the thousands/ten thousands place is only slighty closer to 22/7 than to the 00 of 3.14, so it may give only slightly closer answer than using π. If the problem states what they want you to use, then you have to use that. Most calculators have a π button which will give you the closest answer once you round.

if a square is writen X^2 how is a cube writen

A number cubed, or in this case x, is written as x³, or x^3.

What would be the equation to find the volume for a trapezoid prism?

Any prism volume is V = BH where B is area of base and H is height of prism, so find area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.

Main content

Course: Geometry (all content) > Unit 8

Lesson 4: Volume of cones, cylinders, and spheres

Volume formulas review

Google Classroom

Review the formulas for the volume of prisms, cylinders, pyramids, cones, and spheres.

It may seem at first like there are lots of volume formulas, but many of the formulas share a common structure.

Prisms and prism-like figures

{Volume}_{prism} = (base area) \cdot (height)

We always measure the height of a prism perpendicularly to the plane of its base. That's true even when a prism is on its side or when it tilts (an oblique prism).

Rectangular prisms

Often, we first learn about volume using rectangular prisms (specifically right rectangular prisms), such as by building the prism out of cubes.

Note that any face of a rectangular prism could be its base, as long as we measure the height of the prism perpendicularly to that face.

\begin{aligned} {Volume}_{rectangular prism} & = ({Area}_{rectangle}) \cdot (height) \\ = ((rectangle base) (rectangle height)) \cdot (prism height) \\ = l w h \end{aligned}

Triangular prisms

A triangular prism has a base shaped like a triangle.

\begin{aligned} {Volume}_{triangular prism} & = ({Area}_{triangle}) \cdot (height) \\ = (\frac{1}{2} (triangle base) (triangle height)) \cdot (prism height) \\ = \frac{1}{2} b h ℓ \end{aligned}

Cylinders

A circular cylinder is a prism-like figure that has a base shaped like a circle.

\begin{aligned} {Volume}_{circular cylinder} & = ({Area}_{circle}) \cdot (height) \\ = (π \cdot (radius)^{2}) \cdot (height) \\ = π r^{2} h \end{aligned}

Oblique prisms

In oblique prisms, the bases are in parallel planes,

We still calculate the volume in exactly the same way because of Cavalieri's principle.

Which expression gives the volume of the oblique rectangular prism?

Pyramids and pyramid-like figures

{Volume}_{pyramid} = \frac{1}{3} (base area) \cdot (height)

We also measure the height of a pyramid perpendicularly to the plane of its base. Because of Cavalieri's principle, the same volume formula works for right and oblique pyramid-like figures.

Rectangle-based pyramids

A rectangle-based pyramid has a base shaped like a rectangle.

\begin{aligned} {Volume}_{rectangle-based pyramid} & = \frac{1}{3} ({Area}_{rectangle}) \cdot (height) \\ = \frac{1}{3} ((rectangle base) (rectangle height)) \cdot (pyramid height) \\ = \frac{1}{3} l w h \end{aligned}

Cones

A circular cone is a pyramid-like figure that has a base shaped like a circle.

\begin{aligned} {Volume}_{circular cone} & = \frac{1}{3} ({Area}_{circle}) \cdot (height) \\ = \frac{1}{3} (π \cdot (radius)^{2}) \cdot (height) \\ = \frac{1}{3} π r^{2} h \end{aligned}

Spheres

{Volume}_{sphere} = \frac{4}{3} π (radius)^{3}

We're so glad you asked! Suppose we have a sphere with a radius of

r

. Let's take cross-sections perpendicular to the vertical diameter of the sphere.

By the Pythagorean theorem, the radius

x

of the cross-section of the sphere at a point

y

units from the center is:

x = \sqrt{r^{2} - y^{2}}

So the cross-sectional area of the sphere at that height is:

\begin{aligned} {Area}_{circular cross-section} & = π \cdot x^{2} \\ = π (r^{2} - y^{2}) \end{aligned}

There is another figure that has the same cross-sectional area at every height: a right, circular cylinder with a radius of

r

and a height of

2 r

with two circular cones removed from it. Each cone also has a radius of

r

and has a height of

r

The radius of the cylinder at any height is

r

. By similarity, the radius of the cross-section of each cone at a point

y

units from the apex is

y

So the cross-sectional area of the composed figure at that height is:

\begin{aligned} {Area}_{ring-shaped cross-section} & = π \cdot r^{2} - π \cdot y^{2} \\ = π (r^{2} - y^{2}) \end{aligned}

According to Cavalieri's principle, since the sphere and the composed figure have equal cross-sectional areas at each distance

y

from the center of the figure, the figures also have the same volume.

\begin{aligned} {Volume}_{composed figure} \\ = {Volume}_{cylinder} - 2 \cdot {Volume}_{cone} \\ = [π \cdot (radius)^{2} \cdot (cylinder height)] - 2 \cdot [\frac{1}{3} π \cdot (radius)^{2} \cdot (cone height)] \\ = [π \cdot r^{2} \cdot 2 r] - 2 \cdot [\frac{1}{3} π \cdot r^{2} \cdot r] \\ = 2 π r^{3} - \frac{2}{3} π r^{3} \\ = \frac{4}{3} π r^{3} \end{aligned}

The volume of the composed figure is

\frac{4}{3} π r^{3}

, so by Cavalieri's principle, the volume of the sphere is also

\frac{4}{3} π r^{3}

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ejayan.alghofran
Posted 4 years ago. Direct link to ejayan.alghofran's post “Why is the Volume of sphe...”
Why is the Volume of sphere is divided by 3?
Button navigates to signup pageComment on ejayan.alghofran's post “Why is the Volume of sphe...”
(16 votes)
Answer
- Prisha B
  Posted 4 years ago. Direct link to Prisha B's post “This is a really good vid...”
  This is a really good video that I found:
  https://www.youtube.com/watch?v=xuPl_8o_j7k
  
  It explains the formula with simple geometry and algebra, and in animation.
  Button navigates to signup page
  (26 votes)
hcps-vestc
Posted 7 years ago. Direct link to hcps-vestc's post “if a square is writen X^2...”
if a square is writen X^2 how is a cube writen
Button navigates to signup pageComment on hcps-vestc's post “if a square is writen X^2...”
(0 votes)
Answer
- Ian
  Posted 7 years ago. Direct link to Ian's post “A number cubed, or in thi...”
  A number cubed, or in this case x, is written as x³, or x^3.
  Comment on Ian's post “A number cubed, or in thi...”
  (59 votes)
E.P.I.C.
Posted a year ago. Direct link to E.P.I.C.'s post “who found the proof to th...”
who found the proof to the volume of sphere? better question, how?
Button navigates to signup pageComment on E.P.I.C.'s post “who found the proof to th...”
(9 votes)
Answer
- C4LOwenZ
  Posted 10 months ago. Direct link to C4LOwenZ's post “Click on the line of text...”
  Click on the line of text under the section "Volume of a Sphere" that reads: "Where does this formula come from?" It gives a, in my opinion, somewhat ambiguous answer, but it is an answer nonetheless.
  Button navigates to signup page
  (4 votes)
Eleanor Calandra
Posted 7 years ago. Direct link to Eleanor Calandra's post “When using the pi symbol ...”
When using the pi symbol in calculations, the answer comes out differently than if you just use 3.14, which is how the answers on here are calculated. Why not just recommended to use only 3.14 for pi, not the symbol on calculators as shown on the videos? It is frustrating to get sent back to zero when you use your calculator correctly but you should just use one standard calculation of 3.14.
Button navigates to signup pageComment on Eleanor Calandra's post “When using the pi symbol ...”
(5 votes)
Answer
- Matthew Johnson
  Posted 7 years ago. Direct link to Matthew Johnson's post “3.14 is a simplified vers...”
  3.14 is a simplified version of the real value of pi:
  3.141592653589793238462643383279502884197169399375105820974944592307816406286
  
  http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
  Comment on Matthew Johnson's post “3.14 is a simplified vers...”
  (11 votes)
Marian Oxford
Posted 10 months ago. Direct link to Marian Oxford's post “Will all the volume formu...”
Will all the volume formulas be available on the math sheet at the SAT?
Button navigates to signup pageButton navigates to signup page
(10 votes)
Answer
- hberhane1
  Posted 8 months ago. Direct link to hberhane1's post “prob not lmao”
  prob not lmao
  Comment on hberhane1's post “prob not lmao”
  (0 votes)
AHMED MUHAYMIN
Posted 3 years ago. Direct link to AHMED MUHAYMIN's post “how can I find volume of ...”
how can I find volume of a tetrahedron
Button navigates to signup pageComment on AHMED MUHAYMIN's post “how can I find volume of ...”
(4 votes)
Answer
- kubleeka
  Posted 3 years ago. Direct link to kubleeka's post “A tetrahedron is just a t...”
  A tetrahedron is just a triangular pyramid. If any pyramid has height h and a base of area b, then it's volume is bh/3.
  Button navigates to signup page
  (6 votes)
Elly Murillo
Posted 6 years ago. Direct link to Elly Murillo's post “What would be the equatio...”
What would be the equation to find the volume for a trapezoid prism?
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(0 votes)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “Any prism volume is V = B...”
  Any prism volume is V = BH where B is area of base and H is height of prism, so find area of the base by B = 1/2 h(b1+b2), then multiply by the height of the prism.
  Comment on David Severin's post “Any prism volume is V = B...”
  (14 votes)
Twinmazons
Posted 5 years ago. Direct link to Twinmazons's post “What would be the differe...”
What would be the difference between a triangular prism and a pyramid?
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(3 votes)
Answer
- David Severin
  Posted 5 years ago. Direct link to David Severin's post “A prism has the base that...”
  A prism has the base that goes through the whole figure, so a triangular prism will have two congruent bases that are a triangle. A triangular pyramid has a triangular base and each base vertex that has an edge that goes to a common vertex.
  Comment on David Severin's post “A prism has the base that...”
  (6 votes)
BenjaminD
Posted a year ago. Direct link to BenjaminD's post “01101001 00100000 0110110...”
01101001 00100000 01101100 01101111 01110110 01100101 00100000 01101101 01100001 01111000 00100000 01110000 01100101 01110100 01100101 01110010
Button navigates to signup pageComment on BenjaminD's post “01101001 00100000 0110110...”
(4 votes)
Answer
- cernesamuel
  Posted a year ago. Direct link to cernesamuel's post “01010111 01101000 0110111...”
  01010111 01101000 01101111 00100111 01110011 00100000 01110100 01101000 01100001 01110100 00111111
  Comment on cernesamuel's post “01010111 01101000 0110111...”
  (3 votes)
Purdy AJ not Ryzn
Posted a year ago. Direct link to Purdy AJ not Ryzn's post “When we need to use the v...”
When we need to use the value of pi to find the area of a circle for part of the problem, which number value of pi would be better to use to find a more exact calculation? Would it be 3.14, the symbol for pi, or 22/7?
Button navigates to signup pageComment on Purdy AJ not Ryzn's post “When we need to use the v...”
(3 votes)
Answer
- David Severin
  Posted a year ago. Direct link to David Severin's post “pi = 3.14156... 3.14 22/7...”
  pi = 3.14156...
  3.14
  22/7 = 3.1428...
  So 3.14 will give a slightly underestimate it, and 22/7 will slightly overestimate it. The 15 in the thousands/ten thousands place is only slighty closer to 22/7 than to the 00 of 3.14, so it may give only slightly closer answer than using π. If the problem states what they want you to use, then you have to use that. Most calculators have a π button which will give you the closest answer once you round.
  Button navigates to signup page
  (5 votes)