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13.3 Volumes of spheres. By Kylie Bolton and Jessica Nguyen. Objectives. Find volumes of spheres and hemispheres. Solve problems involving volumes of spheres and hemispheres with given diameters, radius’, surface areas, and circumferences. What is a Sphere?.

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13 3 volumes of spheres

13.3 Volumes of spheres

By Kylie Bolton and Jessica Nguyen

objectives
Objectives
  • Find volumes of spheres and hemispheres.
  • Solve problems involving volumes of spheres and hemispheres with given diameters, radius’, surface areas, and circumferences.
what is a sphere
What is a Sphere?
  • A sphere is the set of points in space equidistant from a certain point.
  • It has no faces, edges, or vertices.
  • An example is our earth.
how do we get the equation for the volume of a sphere
How do we get the equation for the volume of a sphere?

Finding the volume of a sphere is kind of like finding the volume of a pyramid and the surface area of a sphere.

Suppose that the space inside a sphere is divided into an infinite number of near pyramids with all their vertices on the center of the sphere. The height of the pyramid equals the radius of the sphere.

The sum of all the pyramids’ bases will equal the surface area of the entire sphere. The sum of all the pyramids’ volume is equal to the volume of the sphere.

slide5
Recall, V(pyramid)=1/3Bh, where B is the area of its base and h is its height.

Also, the surface area of a sphere, 4∏r², is equal to B1 + B2 + B3 +…+ Bn

Sum of the volumes of all the pyramids.

Replace h with r.

Distributive Property

Replace B1+B2+B3+…+Bn

With 4∏r²

Simplify

V=1/3B¹h¹ + 1/3B²h² + 1/3B³h³ +…+ 1/3Bnhn

=1/3B1r + 1/3B2r + 1/3B3r +…+ 1/3Bnr

=1/3r(B1 +B2 + B3 +…+ Bn)

=1/3r(4∏r²)

=4/3∏r³

volume of a sphere
Volume of a Sphere
  • If a sphere has a volume of a V cubic units and a radius of r units, then V=4/3r³
slide7

Find r

C=2∏r

24=2∏r

R≈3.82m

Plug it in!

V=4/3∏(3.82)³

V=233.5m³

  • Find the Volume of the sphere if C=24m
  • Round to the nearest 10th
example 2
Example 2:
  • Billy was delivering spherical teapots. He accidently spilt the tea out of one. So he needs to know how much tea to put in the teapots.
  • Help him out by finding the volume of the spherical teapot if the radius=12in.
  • Round to the nearest 10th.

12

continued
Continued
  • Find the volume of the jug.
  • V=4/312³
  • V=7238.2in³
  • Substitution
  • Multiply
volume of a hemisphere
Volume of a Hemisphere
  • Hemisphere- half of a sphere.
  • V=1/2(4/3r³)
example
Example:

Find the volume of this hemisphere with a radius of 2in.

2in

continued1
Continued

1. Plug the radius into the formula V=1/2(4/3r³)

2. V=1/2(4/32³)

3. V=1/2(33.5)

4. V=16.8

slide13

Example 1:

Find the volume of the hemisphere if the surface area=10

Round to the nearest 10th

continued2
Continued
  • T=1/2(4r²)+ r²
  • 10=1/2(4r²)+ r²
  • 10=3r²
  • 3.333=r²
  • 1.83=r
  • V=1/2(4/3r³)
  • V=1/2(4/31.83³)
  • V=12.8 units²
  • Surface area of a sphere
  • Substitute
  • Divide both sides by 3∏
  • Find the sq. root
  • Volume of a sphere
  • Substitute
  • Use a calculator
assignment
Assignment

Pg. 704 #3-7, 9-20, 23-24, 30-31

By Jessica and Kylie!