Volume of Cone Calculator
Volume
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Slant Height
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Lateral Surface Area
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Total Surface Area
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How to Calculate the Volume of a Cone
The volume of a cone is exactly one-third the volume of a cylinder with the same base and height: V = (1/3)πr²h. This elegant relationship means a cone holds one-third the liquid of a cylinder with matching dimensions.
The slant height is the distance from the tip to any point on the base edge: l = √(r² + h²). The lateral surface area (curved surface) is LSA = πrl, and the total surface area including the circular base is TSA = πr(r + l).
Cones appear in ice cream cones, traffic pylons, funnels, and architectural structures. Volume calculations help determine capacity, while surface area is useful for material estimation in manufacturing and construction projects.
Frequently Asked Questions
What is the volume of a cone?
V = (1/3)πr²h. A cone with radius 3 and height 12 has volume = (1/3) × π × 9 × 12 ≈ 113.1 cubic units.
How is a cone's volume related to a cylinder?
A cone's volume is exactly one-third of a cylinder with the same radius and height. Three cones fill one cylinder.
What is slant height?
Slant height is the distance from the tip of the cone to the edge of the base, measured along the surface: l = √(r² + h²).
How do you find the surface area of a cone?
Total surface area = πr(r + l), where l is the slant height. This includes the circular base (πr²) plus the lateral surface (πrl).
Why is a cone one-third the volume of a cylinder?
The one-third factor comes from the way a cone tapers from its base to its apex. Mathematically, it can be proven using calculus by integrating the cross-sectional area from base to tip. Intuitively, if you fill three identical cones with water and pour them into a cylinder with the same base radius and height, the cylinder fills exactly. This one-third relationship holds for all cones and pyramids -- any shape that tapers linearly to a point has one-third the volume of the corresponding prism with the same base and height. Archimedes first proved this relationship around 250 BCE.
How do I find the volume of a truncated cone (frustum)?
A frustum is a cone with its top cut off by a plane parallel to the base. Its volume formula is V = (pi/3) times h times (R squared + R times r + r squared), where R is the larger base radius, r is the smaller top radius, and h is the perpendicular height. For example, a frustum with R = 5, r = 3, and h = 8 has volume = (pi/3) times 8 times (25 + 15 + 9) = approximately 411.8 cubic units. Frustums appear in everyday objects like buckets, lampshades, and drinking cups.