Sphere Volume from Cylinder Relationship
Sphere Volume from Cylinder Relationship is a Grade 7 math skill in Big Ideas Math Advanced 2, Chapter 8: Volume and Similar Solids, where students derive and apply the sphere volume formula V = (4/3)*pi*r^3 by understanding that the volume of a sphere equals exactly two-thirds of the volume of the cylinder that circumscribes it. This elegant relationship connects two key volume formulas.
Key Concepts
The volume of a sphere is $\frac{2}{3}$ times the volume of a cylinder that has the same diameter as the sphere and height equal to the diameter: $$V {sphere} = \frac{2}{3} \times V {cylinder} = \frac{2}{3} \times \pi r^2 h$$.
When $h = 2r$ (diameter), this becomes: $$V {sphere} = \frac{2}{3} \times \pi r^2 (2r) = \frac{4}{3}\pi r^3$$.
Common Questions
What is the formula for the volume of a sphere?
V = (4/3) x pi x r^3, where r is the radius of the sphere.
How is the sphere volume related to a cylinder?
A sphere fits exactly inside a cylinder with the same radius and a height equal to the sphere diameter (2r). The sphere volume equals exactly 2/3 of that cylinder volume: V_cylinder = pi x r^2 x 2r = 2*pi*r^3, and (2/3) x 2*pi*r^3 = (4/3)*pi*r^3.
How do you calculate the volume of a sphere with radius 6 cm?
V = (4/3) x 3.14 x 6^3 = (4/3) x 3.14 x 216 = approximately 904.32 cubic cm.
What is Big Ideas Math Advanced 2 Chapter 8 about?
Chapter 8 covers Volume and Similar Solids, including volume formulas for cylinders, cones, and spheres, and exploring relationships between volumes of similar 3D shapes.