Exploring the Equation of a Sphere
Lesson Description
Video Resource
Key Concepts
- Standard equation of a sphere
- Center of a sphere (h, k, j)
- Radius of a sphere (r)
Learning Objectives
- Students will be able to identify the center and radius of a sphere given its equation in standard form.
- Students will be able to write the equation of a sphere in standard form given its center and radius.
Educator Instructions
- Introduction (5 mins)
Begin by reviewing the equation of a circle and its relationship to the equation of a sphere. Emphasize that a sphere is a 3D analog of a circle. - Understanding the Standard Form (10 mins)
Introduce the standard equation of a sphere: (x - h)^2 + (y - k)^2 + (z - j)^2 = r^2. Explain the significance of each variable (h, k, j, r) and their relationship to the center and radius. Point out the importance of the signs (opposite sign for center coordinates). - Example 1: Writing the Equation (10 mins)
Work through an example where students are given the center and radius and asked to write the equation of the sphere. Follow the example in the video: r = 3, C(-2, 1, 4). Emphasize the substitution process and simplification. - Example 2: Identifying Center and Radius (10 mins)
Work through an example where students are given the equation of a sphere and asked to identify the center and radius. Follow the example in the video: (x + 3)^2 + y^2 + (z - 7)^2 = 100. Emphasize extracting the center coordinates with correct signs and taking the square root to find the radius. - Practice Problems (10 mins)
Provide students with additional practice problems to work on independently or in small groups. Include problems that require both writing the equation and identifying the center and radius.
Interactive Exercises
- Equation Matching
Provide a list of sphere equations and a list of center/radius coordinates. Have students match each equation to its corresponding center and radius. - Graph Visualisation
Use graphing software (e.g. GeoGebra) to plot spheres given their equations. This will help students visualise the relationship between the equation and the 3D shape.
Discussion Questions
- How does the equation of a sphere relate to the equation of a circle?
- Why do we take the square root of the right side of the equation to find the radius?
- What happens to the equation if the center of the sphere is at the origin?
Skills Developed
- Algebraic manipulation
- Spatial reasoning
- Equation interpretation
Multiple Choice Questions
Question 1:
The standard form equation of a sphere is (x - h)^2 + (y - k)^2 + (z - j)^2 = r^2. What does 'r' represent?
Correct Answer: Radius
Question 2:
The center of the sphere is at (-1, 0, 2), and the radius is 5. Which of the following is the equation of the sphere?
Correct Answer: (x + 1)^2 + y^2 + (z - 2)^2 = 25
Question 3:
What is the center of the sphere defined by the equation (x + 4)^2 + (y - 3)^2 + z^2 = 16?
Correct Answer: (-4, 3, 0)
Question 4:
The equation of a sphere is (x - 2)^2 + (y + 1)^2 + (z - 5)^2 = 9. What is the radius of the sphere?
Correct Answer: 3
Question 5:
Which of the following points lies on the sphere with the equation x^2 + y^2 + z^2 = 1?
Correct Answer: (1, 0, 0)
Question 6:
The diameter of a sphere is 12. Its equation is centered at the origin. Which of the following represents this sphere?
Correct Answer: x^2 + y^2 + z^2 = 36
Question 7:
Given the equation of a sphere (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, what happens to the sphere if 'r' is doubled?
Correct Answer: The radius is doubled
Question 8:
A sphere has its center on the z-axis at (0, 0, 5) and a radius of 3. What is the equation of this sphere?
Correct Answer: x^2 + y^2 + (z-5)^2 = 9
Question 9:
If the center of a sphere is at the origin (0,0,0) and passes through the point (1,2,2), what is the sphere's radius?
Correct Answer: 3
Question 10:
The equation of a sphere is given as (x-1)^2 + (y+2)^2 + (z-3)^2 = k. If the sphere passes through the point (1, -2, 6), what is the value of k?
Correct Answer: 36
Fill in the Blank Questions
Question 1:
The standard equation of a sphere is (x - h)^2 + (y - k)^2 + (z - j)^2 = ____^2.
Correct Answer: r
Question 2:
In the equation (x + 2)^2 + (y - 3)^2 + (z - 1)^2 = 4, the x-coordinate of the center of the sphere is ____.
Correct Answer: -2
Question 3:
Given the equation of a sphere (x - 5)^2 + y^2 + (z + 4)^2 = 16, the radius of the sphere is ____.
Correct Answer: 4
Question 4:
If a sphere's center is at the origin (0, 0, 0) and its radius is 7, the equation of the sphere is x^2 + y^2 + z^2 = ____.
Correct Answer: 49
Question 5:
If the equation of a sphere is (x+1)^2 + (y-5)^2 + (z+3)^2 = 25, the y-coordinate of the sphere's center is ____.
Correct Answer: 5
Question 6:
The equation of a sphere with a center at (2, -1, 0) and a radius of √5 is (x - 2)^2 + (y + 1)^2 + z^2 = ____.
Correct Answer: 5
Question 7:
A sphere centered at (1, 1, 1) that is tangent to the xy-plane has a radius of ____.
Correct Answer: 1
Question 8:
For a sphere defined by x^2 + (y - 3)^2 + (z + 2)^2 = 9, the z-coordinate of the center is ____.
Correct Answer: -2
Question 9:
The diameter of a sphere is twice the length of its ____.
Correct Answer: radius
Question 10:
Given the equation (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, the distance from the center of the sphere to any point on its surface is equal to ____.
Correct Answer: r
Educational Standards
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