Equations of Spheres
Presentation
•
Mathematics
•
10th - 12th Grade
•
Easy
Bo Anthony
Used 2+ times
FREE Resource
7 Slides • 5 Questions
1
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2
When plotting a point in space, we need a 3rd Axis, the z-axis
Because of this, any point in space would have the coordinate of (x,y,z) ALWAYS IN THIS ORDER!!
We can use the same midpoint and distance formulas from Algebra for these 3D points, just with the z-portion added on
3
(x-h)2 + (y-k)2 + (z-l)2 = r2
Center located at ( h, k, l ) with Radius r.
4
What would be the equation of the Sphere with a Center at (1, -2, 4) and a Radius of 2?
Plug the h, k, l, and r into the Sphere Equation: (x - h)2 + (y - k)2 + (z - l)2 = r2
(x - 1)2 + (y - -2)2 + (z - 4)2 = 22
Simplify (if needed)
(x - 1)2 + (y + 2)2 + (z - 4)2 = 4
5
Multiple Choice
Example 2: Which of the following would be the correct Equation for a Sphere with a Center at (-1, 0, 3) and a Radius of 10?
(x + 1)2 + y2 + (z - 3)2 = 100
(x - 1)2 + y2 + (z + 3)2 = 100
(z + 1)2 - (x + 3)2 - y2 = 100
6
Multiple Choice
Example 3: What would be the equation for a Circle with a center of (8,3,7) and a Radius of 14?
(x - 3)2 + (y - 8)2 + (z - 7)2 = 196
(x - 8)2 + (y - 3)2 + (z - 7)2 = 196
(x - 4)2 + (y - 1.5)2 + (z - 3.5)2 = 196
7
What if it isn't in Standard Form?
If you are given a Sphere equation that ISN'T in Standard Form, to convert it to that, you'll need to go through the steps of Completing the Square, for ALL 3 VARIABLES!!!
The next slide has a video example going through this to help remind you of the steps for Completing the Square...
8
Converting an Equation to Standard Form
9
Converting an Equation to Standard Form (2nd Ex)
10
Multiple Choice
Now you give it a Try:
Convert to the Standard Form of a Sphere:
x2+y2+z2−8x+10y+20z−8=0
(x−4)2+(y+5)2+(z+10)2=149
(x−8)2+(y+10)2+(z+20)2=149
(x−8)2+(y+10)2+(z+20)2=8
(x−4)2+(y+5)2+(z+10)2=8
11
Multiple Choice
Example 2: Convert to Standard Form
y2+x2+z2+5y−7x+8z=4
(x−3.5)2+(y+2.5)2+(z+4)2=14
(x−3.5)2+(y+2.5)2+(z+4)2=38.5
(x−3.5)2+(y+2.5)2+(z+4)2=4
(x−7)2+(y+5)2+(z+8)2=38.5
12
Draw
Summary: Draw a Cat
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