Calculate the volume, surface area, lateral area, and slant height of a cone. Supports multiple solve modes, unit conversion, and real-world presets.
A cone is one of the most recognizable three-dimensional shapes, featuring a circular base that tapers smoothly to a single point called the apex. Cones appear everywhere in daily life — from ice cream cones and traffic cones to party hats and architectural spires. Understanding how to calculate the volume and surface area of a cone is a fundamental skill in geometry, engineering, and manufacturing.
The volume of a cone equals one-third the volume of a cylinder with the same base and height, expressed as V = ⅓πr²h. This elegant relationship means a cone holds exactly one-third the material of its cylindrical counterpart. The total surface area combines the circular base (πr²) with the curved lateral surface (πrl), where l is the slant height calculated using the Pythagorean theorem: l = √(r² + h²).
This calculator supports three solve modes: given radius and height, given radius and slant height, or reverse-solving height from a known volume and radius. It also includes presets for common real-world cones, a surface-area breakdown with visual ratio bars, and a reference table so you can quickly compare dimensions of familiar conical objects.
This cone volume calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Solve Mode, Units, Radius, Height and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Volume: V = ⅓πr²h Lateral Surface Area: A_l = πrl Base Area: A_b = πr² Total Surface Area: A = πr(r + l) Slant Height: l = √(r² + h²)
Result: For solvemode=5, units=10, radius=15, the tool returns the solved cone volume outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in cone volume formulas and reports derived values, checks, and classifications automatically.
This page is tailored to cone volume, with outputs tied directly to the form fields (Solve Mode, Units, Radius, Height). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
The volume of a cone is V = ⅓πr²h, where r is the base radius and h is the perpendicular height from the base to the apex. Use this as a practical reminder before finalizing the result.
Use the Pythagorean theorem: slant height l = √(r² + h²), since the radius, height, and slant height form a right triangle. Keep this note short and outcome-focused for reuse.
Lateral area is the curved side surface only (πrl). Total surface area adds the flat circular base (πr²), giving πr(r + l).
This can be proven with calculus (integration of circular cross-sections) or Cavalieri's principle. Three cones fit exactly inside a cylinder of the same base and height.
Yes — first compute the height: h = √(l² − r²), then use V = ⅓πr²h. This calculator's "Radius + Slant Height" mode does this automatically.
1 liter = 1,000 cm³ and 1 US gallon ≈ 3,785.41 cm³. Calculate volume in cm³ first, then divide accordingly.