1. 12.6 Surface Area and Volume of Spheres
Unit V Day 6

2. Do Now List the formulas: Euler’s theorem Volume of prism
Volume of cylinder
Surface area of prism
Surface area of cylinder
Volume of pyramid
Volume of cone

3. Finding the Surface Area of a Sphere
We know that a circle is the set of all points in a plane that are a given distance from a point.
A sphere is the set of all points in space that are a given distance (radius) from a given point (center).

4. Ex. 1: Finding the Surface Area of a Sphere (baseball)
A baseball and its leather covering are shown. The baseball has a radius of about 1.45 inches.
Estimate the amount of leather used to cover the baseball.
a. S = 4π(1.45)2 ≈ 26.4º

5. Theorem 12.11: Surface Area of a Sphere
The surface area of a sphere with radius r is S = 4r2.

6. Ex. 1: Finding the Surface Area of a Sphere
Find the surface area of each sphere.
When the radius doubles, does the surface area double?
S = 4π*22 = 16π in.2
S = 4π*42 = 64π in.2
The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16π • 4 = 64π.
So, when the radius of a sphere doubles, the surface area does not double.
Note: It appears to quadruple, but this is not a proof of that fact. To prove it, we would need to use radii of r and 2r.

7. Finding the Volume of a Sphere
Let n be the number of pyramids needed to fill the sphere.
The volume of each pyramid is __________.
The sum of the base areas is _____. This is the __________________, which is ______.
You can approximate the volume V of the sphere as _____________.
Imagine that the interior of a sphere with radius r is approximated by n pyramids as shown, each with a base area of B and a height of r, as shown.
1/3 Br
nB; surface area; 4πr2
Volume of the sphere ≈ n*volume of pyramid = n*1/3 Br
Since nB ≈ 4πr2, we can substitute to get V ≈ 1/3(4πr2)r = 4/3 r3

8. Theorem 12.12: Volume of a Sphere
The volume of a sphere with radius r is S = 4/3 r3. 3

9. Ex. 4: Finding the Volume of a Sphere (ball bearings)
To make a steel ball bearing, a cylindrical slug is heated and pressed into a spherical shape with the same volume. Find the radius of the ball bearing to the right.
To find the volume of the slug, use the formula for the volume of a cylinder.
V = r2h
= (12)(2)
= 2 cm3
To find the radius of the ball bearing, use the formula for the volume of a sphere and solve for r.
V =4/3r3
2 = 4/3r3
6 = 4r3
1.5 = r3
1.14  r

10. Closure
How can you find the surface area of a sphere? the volume of a sphere?
Use the formulas S= 4πr2 and V = 4/3πr3