What This Calculator Does
You need to figure out how much material it takes to make a funnel, or how much water a conical tank can hold. This cone calculator takes a radius and a height and gives you the volume, slant height, lateral surface area, total surface area, and base area in one pass. It works for right circular cones, which is the shape most people picture when they hear the word cone: a circular base that tapers up to a single point directly above the center.
The tool also shows every formula with your numbers substituted in, so you can trace how each result was reached. That matters for students checking homework and for engineers who need to verify a figure before ordering material. For the flat circle that forms the base, you can cross-check with our Circle Calculator.
Inputs Required
- Radius (r): The radius of the circular base
- Height (h): The perpendicular distance from the base to the apex
Outputs Provided
- Volume: The space inside the cone
- Slant Height: The distance from the apex to any point on the base circle's edge
- Lateral Surface Area: The area of the curved side only
- Total Surface Area: The curved side plus the circular base
- Base Area: The area of the circular base
How the Calculation Works
A right circular cone has three defining measurements: the base radius r, the perpendicular height h, and the slant height l. The slant height is the hypotenuse of the right triangle formed by the radius and the height, so it comes straight from the Pythagorean theorem.
Slant Height: l = √(r² + h²)
Volume: V = (1/3)πr²h
Lateral Surface Area: LSA = πrl
Total Surface Area: SA = πr² + πrl = πr(r + l)
Base Area: A = πr²
The volume formula is one third of the cylinder volume for the same radius and height. That ratio comes from calculus: integrating circular cross-sections that shrink as you move toward the apex gives exactly one third of the cylinder result. Archimedes first proved this relationship, and he was so proud of it that he asked for a sphere inscribed in a cylinder to be carved on his tombstone. The lateral surface area equals πrl because unrolling the curved side produces a sector of a circle with radius l and arc length 2πr. For general 3D volume work across multiple shapes, see our Volume Calculator.
How to Use the Calculator
- Enter the radius of the cone's circular base. Use any unit (inches, centimeters, feet) as long as you use the same unit for height.
- Enter the perpendicular height, measured straight up from the base center to the apex. Do not enter the slant height here.
- Read the volume, slant height, and surface areas from the results panel. The formulas with substituted values appear below the result cards.
- Use the Copy Result button to save the full output to your clipboard.
Example Calculations
Example 1: A Conical Water Tank
A municipality installs an inverted cone tank with a radius of 5 feet and a height of 12 feet. The slant height is √(25 + 144) = √169 = 13 feet. The volume is (1/3)π × 25 × 12 = 100π ≈ 314.16 cubic feet, which holds about 2,351 gallons. The total surface area is π × 5 × (5 + 13) = 90π ≈ 282.74 square feet, which tells the fabrication shop how much steel sheet to order for the cone wall and base.
Example 2: A Paper Party Hat
A manufacturer designs a party hat with a base radius of 3 inches and a height of 8 inches. The slant height is √(9 + 64) = √73 ≈ 8.54 inches. The lateral surface area is π × 3 × 8.54 ≈ 80.52 square inches, which is the amount of cardstock needed per hat. The volume is (1/3)π × 9 × 8 ≈ 75.40 cubic inches, useful for estimating how much confetti the hat can hold before it tips over.
Real World Scenarios
Sizing a Traffic Cone for Wind Load
A transportation engineer specifies traffic cones for a highway project. A standard 28-inch cone has a base radius of about 7 inches and a height of 28 inches. The lateral surface area of roughly π × 7 × √(49 + 784) ≈ 631 square inches tells the engineer how much wind the cone catches. Combined with the cone's weight, this determines whether the cone stays upright in a 40 mph gust or needs a rubber base weight added.
Estimating Concrete for a Conical Pier Footing
A contractor pours a conical footing with a 2-foot radius and 3-foot depth. The volume is (1/3)π × 4 × 3 = 4π ≈ 12.57 cubic feet, or about 0.47 cubic yards. Since ready-mix trucks sell in quarter-yard increments, the contractor orders half a yard for this footing plus a margin for over-excavation. Getting the volume wrong by ignoring the 1/3 factor would double the order and waste money.
Designing a Funnels for a Food Processing Line
A food plant engineer designs a stainless steel funnel with a 10-inch radius and 14-inch height. The slant height is √(100 + 196) ≈ 17.20 inches. The lateral surface area of π × 10 × 17.20 ≈ 540.35 square inches determines how much sheet metal to cut. Because the funnel wall is a flattened sector of a circle, the engineer also needs the arc length (2πr = 62.83 inches) and the sector radius (the slant height, 17.20 inches) to lay out the flat pattern before rolling.
Common Mistakes to Avoid
- Confusing slant height with perpendicular height: The volume formula uses the perpendicular height h, not the slant height l. If you only know the slant height and the radius, compute h = √(l² - r²) first.
- Forgetting the 1/3 factor in volume: A cone holds one third as much as a cylinder with the same base and height. Dropping the 1/3 overstates the volume by 200%, which leads to ordering too much material or too large a tank.
- Mixing units between radius and height: If the radius is in inches and the height is in feet, the volume comes out in a meaningless mixed unit. Convert both to the same unit before entering them.
- Using lateral area when you need total area: Lateral surface area excludes the base. If you are pricing material for a closed cone (like a tank with a bottom), add the base area πr² to get the total.
Limitations of This Calculator
This tool handles right circular cones only, where the apex sits directly above the center of the base. It does not work for oblique cones (where the apex is off-center), truncated cones or frustums, or elliptical cones. For frustum volume, use the formula V = (1/3)πh(R² + Rr + r²) with the two radii. The calculator also assumes you are entering the perpendicular height, not the slant height.
Authoritative Research and Resources
- Wolfram MathWorld: Cone - A detailed reference covering the geometry of cones, including volume and surface area derivations, with the formal calculus behind the 1/3 factor.
- Maths Is Fun: Cone - An accessible explanation of cone properties with interactive diagrams, useful for students who want a visual breakdown of the slant height relationship.
For related tools, calculate cylinder properties with our Cylinder Calculator, sphere properties with our Sphere Calculator, or surface area across multiple shapes with our Surface Area Calculator.