Volume of a Sphere and Cone
A sphere with radius r has exactly twice the volume of a cone with the same radius and height equal to the sphere diameter (h = 2r). This relationship derives from substituting h = 2r into the cone formula: V cone = (1/3) pi r squared (2r) = (2/3) pi r cubed, and V sphere = 2 times (2/3) pi r cubed = (4/3) pi r cubed. A cone with r = 3 cm and h = 6 cm has volume 18 pi cm cubed; the matching sphere has volume 36 pi cm cubed = 2 times 18 pi. This elegant relationship from enVision Mathematics, Grade 8, Chapter 8 deepens 8th grade understanding of 3D volume formulas.
Key Concepts
The volume of a sphere is exactly twice the volume of a cone that has the same radius ($r$) and a height ($h$) equal to the sphere's diameter ($2r$).
$$V {\text{sphere}} = 2 \cdot V {\text{cone}} \quad \text{when } h = 2r$$.
Common Questions
How does the sphere volume relate to a cone with the same radius?
A sphere has exactly twice the volume of a cone that shares its radius and has a height equal to the sphere diameter (h = 2r). V sphere = 2 times V cone in this specific case.
What is the volume formula for a sphere?
V sphere = (4/3) pi r cubed, where r is the radius.
What is the volume formula for a cone?
V cone = (1/3) pi r squared h, where r is the base radius and h is the perpendicular height.
A cone has r = 5 cm and h = 10 cm. What is the volume of a matching sphere with r = 5 cm?
V cone = (1/3) pi (25)(10) = 250 pi/3 cm cubed. V sphere = 2 times V cone = 500 pi/3 cm cubed. Or directly: V sphere = (4/3) pi (125) = 500 pi/3 cm cubed.
Why does the cone height need to equal 2r for this relationship to hold?
Only when h = 2r does the cone volume simplify to (2/3) pi r cubed, which is exactly half of the sphere formula (4/3) pi r cubed.
When do 8th graders learn about sphere and cone volume relationships?
Chapter 8 of enVision Mathematics, Grade 8 covers this in the Surface Area and Volume unit.