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? 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?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.

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

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

Sum of the volumes of all the pyramids.

Replace h with r.

Distributive Property

Replace B1+B2+B3+…+Bn With 4∏r²

Simplify

Volume of a Sphere

If a sphere has a volume of a V cubic units and a radius of r units, then V=4/3r³

Find the Volume of the sphere if C=24m Round to the nearest 10th

Find rC=2∏r24=2∏rR≈3.82m

Plug it in!V=4/3∏(3.82)³V=233.5m³

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

Find the volume of the jug.

V=4/312³ V=7238.2in³

Substitution Multiply

Volume of a Hemisphere

Hemisphere- half of a sphere.

V=1/2(4/3r³)

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

2in

Continued1. Plug the radius into the formula V=1/2(4/3r³)2. V=1/2(4/32³)3. V=1/2(33.5)4. V=16.8

FIND THE VOLUME OF THE HEMISPHERE IF THE SURFACE AREA=10ROUND TO THE NEAREST 10TH

Example 1:

Continued T=1/2(4r²)+ r² 10=1/2(4r²)+ r² 10=3r² 3.333=r² 1.83=r V=1/2(4/3r³) V=1/2(4/31.83³) 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

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

By Jessica and Kylie!