Volume of Cones – V = ⅓πr²h
A cone's volume is exactly ⅓ of a cylinder's volume with the same base radius and height: V = ⅓πr²h. This exact one-third relationship isn't a coincidence or an approximation – it can be proven rigorously, and it holds for pyramids and their matching prisms too (a pyramid is always ⅓ of the prism sharing its base and height).
A simple way to feel this relationship: if you filled an empty cone with water and poured it into a cylinder with the same base and height, it would take exactly 3 full cones to fill the cylinder to the top – a classic hands-on demonstration used in classrooms for generations.
Calculating a Cone's Volume
V = ⅓πr²h. A cone's volume is always exactly ⅓ of the matching cylinder's volume (same r and h).
V = ⅓π(3)²(7) = ⅓π(9)(7) = 21π ≈ 65.97 cm³.
Real-Life Application
- Ice cream cones: calculating how much ice cream fits inside a cone.
- Funnels: estimating the capacity of a cone-shaped funnel.
- Volcanoes: estimating the volume of a cone-shaped volcanic mound.
Key Takeaways
- A cone's volume formula is V = ⅓πr²h.
- A cone's volume is always exactly ⅓ of the matching cylinder's volume.
- The same ⅓ relationship holds between any pyramid and its matching prism.
Practice: Volume of Cones
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