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13-3 Volume of Spheres. Formula for Finding the Volume of a Sphere. 4/3. W a i t f o r i t. Why You Ask?!!. Imagine that we have a disco ball. Now imagine that we cut one of the squares (mirrors) of the disco ball out all the way to the center, narrowing down until it comes to a point.

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Why you ask
Why You Ask?!!

  • Imagine that we have a disco ball.

  • Now imagine that we cut one of the squares (mirrors) of the disco ball out all the way to the center, narrowing down until it comes to a point.

  • We now have a pyramid.

  • Just like a sphere has an infinite amount of great circles, they have an infinite amount of these pyramids.

Height=Radius of Sphere


Why does the height of the pyramid the radius of the sphere
Why Does the Height of the Pyramid=The Radius of the Sphere?

l

  • The lateral height of the pyramid cannot be the radius of the sphere because we cannot solve for an actual sphere. Instead we search for a shape that is close to a sphere (like a disco ball) and solve for that. If we try to solve for an actual sphere, we wouldn’t have a pyramid, but a curved shape, which isn’t a pyramid.

  • This shape has a base that curves inward with the surface of the sphere. With a pyramid, which we CAN solve for, the lateral height would actually be longer than the radius. The height of the pyramid would be the only segment that goes from the center of the sphere to the surface of the sphere. So, we get the radius. THE END

h

l

h


Back to the equation
Back to the equation

  • V=1/3B¹h¹+B²h²+B³h³+. . . +Bªhª

  • V=1/3B¹r+B²r+B³r+. . . +Bªr

  • V=1/3r(B¹+B²+B³+. . . +Bª)

  • V=1/3r(4)

  • V=4/3

  • Volume of Infinite Pyramids

  • H can be replaced with the radius

  • Distributive Property

  • All the bases added together would be the surface area, which is 4

  • Simplify


So lets try it!Find the volume of the sphere

8cm


Steps
Steps

V= 4/3r³

4/3(8)³

4/3(512)

4/3(1608.4954)

6433.9818/3

2144.6606

V=2,144.7cm³


Your turn!!

  • Find the volume of a sphere with a diameter of 22m.

22m


Steps1
Steps

  • V= 4/3r³

  • 4/3(11)³

  • 4/3(1331)

  • 4/3(4181.4598)

  • 16725.8393/3

  • 5575.2798

  • V=5575.3m³


More complications
More Complications!! :)

  • Find the volume of the hemisphere with a radius of 5 cm

5 cm


Steps2
Steps

  • 1/2(4/3)

  • 1/2(4/3)

  • 1/2(4/3125)

  • 1/2(166.6667)

  • 1/2(523.5988)

  • V=261.8


Your turn
Your turn!!

  • Find the volume of a hemisphere with a diameter of 8 inches.

8 in


Steps3
Steps

  • 1/2(4/3)

  • 1/2(4/3)

  • 1/2(4/364)

  • 1/2(85.3333)

  • 1/2(268.0825)

  • V=134


Here s a nasty one well not really

Here’s a nasty one!... Well, not really...

C=37.68m


Steps4
Steps 37.68 m. Use 3.14 for

  • 37.68

  • 37.68/

  • r=6

  • 1/2(4/3)

  • 1/2(4/3216)

  • 1/2(288)

  • 1/2(904.7786842)

  • V=452.4


Assignment
Assignment 37.68 m. Use 3.14 for

  • Pre-AP pg. 704 9-22, 30, 31

Have so much fun!!!


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