Relationship Between Cone and Cylinder Volumes
The relationship between cone and cylinder volumes is a Grade 7 geometry concept in Big Ideas Math Advanced 2, Chapter 8: Volume and Similar Solids. A cone has exactly one-third the volume of a cylinder with the same base and height, which is why the cone volume formula includes the factor one-third. If a cylinder holds 300 mL, a cone with identical dimensions holds only 100 mL.
Key Concepts
A cone has exactly one third the volume of a cylinder with the same base and height: $V {cone} = \frac{1}{3} \cdot V {cylinder}$.
If cylinder volume is $V {cylinder} = \pi r^2 h$, then cone volume is $V {cone} = \frac{1}{3}\pi r^2 h$.
Common Questions
What is the relationship between cone and cylinder volumes?
A cone with the same base and height as a cylinder holds exactly one-third of the cylinder volume. This is why the cone volume formula is one-third times pi r squared times h.
Why does the cone volume formula have one-third in it?
The one-third factor comes from the fundamental geometric relationship: a cone always occupies exactly one-third of the space of a cylinder with the same base and height.
How can you use cylinder volume to find cone volume?
Find the cylinder volume using pi r squared h, then divide by 3 to get the cone volume.
What textbook covers the cone and cylinder volume relationship in Grade 7?
Big Ideas Math Advanced 2, Chapter 8: Volume and Similar Solids covers the one-third relationship between cone and cylinder volumes.