Cones and Spheres

Examples, solutions and videos to help Grade 8 students learn how to use the Pythagorean Theorem to determine an unknown dimension of a cone or a sphere.

New York State Common Core Math Grade 8, Module 7, Lesson 19

Worksheets for Grade 8

Lesson 19 Student Outcomes

• Students use the Pythagorean Theorem to determine an unknown dimension of a cone or a sphere.
  
• Students know that a pyramid is a special type of cone with triangular faces and a rectangular base.
  = Students know how to use the lateral length of a cone and the length of a chord of a sphere to solve problems related to volume.

Lesson 19 Summary

The volume formula for a right square pyramid is 1/3 Bh, where B is the area of the square base.
Given the lateral length and the length of the radius, the Pythagorean Theorem can be used to determine the height of the cone.

Lesson 19 Classwork

Opening Exercises
Note: Figures not drawn to scale.

1. Determine the volume for each figure below.
   a. Write an expression that shows volume in terms of the area of the base, B, and the height of the figure. Explain the meaning of the expression, and then use it to determine the volume of the figure.
   b. Write an expression that shows volume in terms of the area of the base, B, and the height of the figure. Explain the meaning of the expression, and then use it to determine the volume of the figure.
2. a. Write an expression that shows volume in terms of the area of the base, B, and the height of the figure. Explain the meaning of the expression, and then use it to determine the volume of the figure.
   b. The volume of the pyramid shown below is 480 in³. What do you think the formula to find the volume of pyramid is? Explain your reasoning.

Example 1
State as many facts as you can about a cone.
Exercises
3. What is the lateral length of the cone shown below?
4. Determine the exact volume of the cone shown below.
5. What is the lateral length (slant height) of the pyramid shown below? Give an exact square root answer and an approximate answer rounded to the tenths place.
6. Determine the volume of the pyramid shown below. Give an exact answer using a square root.
7. What is the length of the chord of the sphere shown below? Give an exact answer using a square root.
8. What is the length of the chord of the sphere shown below? Give an exact answer using a square root.
9. What is the volume of the sphere shown below? Give an exact answer using a square root.
10. What is the volume of the sphere shown below? Give an exact answer using a square root.

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