This cone calculator handles a right circular cone from either pair of measurements you are likely to have. Give it a radius or a diameter, then either the perpendicular height or the slant height, and it returns volume, the missing height, lateral surface area, total surface area, base area and capacity in litres and US gallons. Every result is shown with the substitution line, so you can see that a cone with radius 4 cm and height 9 cm gives V = ⅓ × π × 4² × 9 = 150.80 cm³.
Arb Digital keeps the two surface areas separate on purpose. A cone has a curved side and a flat circular base, and roughly half the practical questions people ask about cones concern only one of them. Merging them into a single number is the mistake that makes most cone pages useless for real work.
What This Cone Calculator Does
The calculator solves the full property set of a right circular cone, meaning a cone whose apex sits directly above the centre of its circular base. You choose whether your base figure is a radius or a diameter, because tubes, funnels, moulds and pipe fittings are almost always specified by diameter while textbooks almost always use radius. Selecting diameter simply halves the value before anything else happens.
You then choose whether your second measurement is the perpendicular height, measured straight up from the centre of the base to the apex, or the slant height, measured along the sloping surface from the rim to the apex. Those two are not the same number and mixing them up is the single most common cone error. If you give one, the calculator derives the other using the right triangle formed by the radius, the height and the slant.
The results split surface area into three parts: the curved lateral area, the flat base area, and their sum as total surface area. Volume appears as the headline number, with capacity in litres and gallons underneath, which is the figure you need for funnels, hoppers and conical tanks.
How to Use It
- Choose radius or diameter and enter the base measurement. The label updates so there is no ambiguity about which one the field expects.
- Choose height or slant height and enter that value. Use whichever you physically measured rather than converting by hand.
- Set the unit. All results come back in that unit, squared for areas and cubed for volume.
- Read the volume in the hero panel and the four supporting figures in the tiles beneath it.
- Check the working line to confirm the substitution matches the numbers you intended to enter.
The Formula: How a Cone Is Calculated
With radius r, perpendicular height h and slant height l:
- Volume: V = ⅓πr²h. With r = 4 cm and h = 9 cm: ⅓ × π × 16 × 9 = 48π = 150.80 cm³, or about 0.151 litres.
- Slant height: l = √(r² + h²). Here √(16 + 81) = √97 = 9.849 cm.
- Lateral (curved) surface area: Aₗ = πrl. That is π × 4 × 9.849 = 123.79 cm².
- Base area: Aₖ = πr² = 50.27 cm².
- Total surface area: A = πr(r + l) = πr² + πrl = 174.06 cm².
The slant height relation is pure Pythagoras. Slice the cone vertically through the apex and you get an isosceles triangle; half of it is a right triangle whose legs are the radius and the perpendicular height, with the slant height as the hypotenuse. Going the other way, if the slant height is what you measured, then h = √(l² − r²), which requires l to be strictly greater than r. The full set of cone identities is catalogued at Wolfram MathWorld's cone entry.
Lateral Area vs Total Area: Which One You Actually Need
Lateral surface area is the curved sloping face alone. Total surface area adds the flat circular base. For the worked example the difference is 123.79 cm² against 174.06 cm², so choosing the wrong one is a 41% error.
Use lateral area when the base is open or not part of the surface: a paper party hat, a traffic cone, a lampshade, a funnel, a conical roof, the label wrapped around a cone-shaped cup. Use total area when the solid is closed and every face counts: a solid cone being painted or plated, a sealed conical container, a cast component.
There is a neat way to picture lateral area. Cut a cone up the side and flatten it, and you get a circular sector of radius l whose arc length equals the base circumference 2πr. The area of that sector is πrl, which is exactly the lateral area formula. That unrolled sector is also the template you cut if you are making a cone out of sheet material, so the number does double duty as both an area and a pattern dimension. For the flat circle side of the problem, our circle calculator handles radius, circumference, area and sector arithmetic.
Why the Cone Is Exactly One Third of Its Cylinder
The ⅓ in the volume formula is not a fudge factor. A cone and a cylinder that share the same base radius and the same height stand in an exact 1:3 ratio: the cone holds precisely one third as much. The relationship holds for any radius and any height, and it is one of the oldest results in solid geometry.
That gives you a fast sanity check with no calculator at all. Work out the cylinder volume πr²h, then divide by three. For r = 4 cm and h = 9 cm the cylinder holds 452.39 cm³, so the cone holds 150.80 cm³. If your cone answer is not close to a third of the matching cylinder, you have made an arithmetic error somewhere. You can run the cylinder side of that check with the cylinder volume calculator.
The same one-third factor applies to pyramids: any cone or pyramid is one third of the prism or cylinder with the same base and height. It is a property of tapering to a point, not of being round.
Conical Tanks, Funnels and Partial Fill
A conical hopper or funnel does not fill in a way that matches intuition. Because volume depends on the square of the radius at any given level, the top of a point-down cone holds far more than the bottom. Fill a point-down cone to half its height and you capture only one eighth of its total volume, since both the radius and the depth of the filled portion are halved and the volume scales with the cube of the linear scale factor.
That means a sight gauge marked evenly up the side of a conical vessel is badly misleading. The bottom half of the height holds 12.5% of the contents and the top half holds 87.5%. Anyone reading depth as if it were proportional to volume will systematically overestimate what is left in a draining hopper. The correct partial volume for a point-down cone filled to depth d, where the full cone has height h, is the full volume multiplied by (d/h)³.
Capacity conversion is the other half of the job. A cubic centimetre is exactly one millilitre, so cubic centimetres divide by 1,000 to give litres, and litres divide by 3.785411784 to give US gallons. If you need to move an existing volume figure between units without recalculating any geometry, the volume converter is the right tool.
Right Cones, Oblique Cones and What This Tool Assumes
This calculator assumes a right circular cone. If the apex is offset so it does not sit above the centre of the base, the solid is an oblique cone. Its volume formula is unchanged, still ⅓πr²h with h measured as the perpendicular distance from base plane to apex, but its surface area is not πrl and has no simple closed form. Do not use the lateral area figure from this page for an oblique cone.
A truncated cone, or frustum, is a cone with its tip cut off parallel to the base. Its volume is the whole cone minus the removed tip, and it needs both the top and bottom radii to compute. Buckets, plant pots and most drinking cups are frustums rather than true cones, so check the shape before you trust a cone formula on a real object. Unit definitions used throughout this page follow NIST's guide to the SI.
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Browse All Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Using slant height in the volume formula. Volume needs the perpendicular height. Substituting the slant height inflates the answer every time.
- Forgetting the one-third factor. πr²h is the cylinder. The cone is a third of that.
- Entering a diameter as a radius. That doubles r, and since volume depends on r², the result comes out four times too large.
- Adding the base when the base is open. Party hats, funnels and traffic cones need lateral area only.
- Reading a conical tank level as if it were linear. Half the height of a point-down cone is only one eighth of the volume.
Related Free Tools From Arb Digital
Compare against the cylinder volume calculator to see the one-third relationship directly, or the sphere volume calculator for the other classic curved solid. The circle calculator covers the base on its own, and the Pythagorean theorem calculator shows the right-triangle step that produces slant height. Browse the full free online tools hub for everything else.
Frequently Asked Questions
Volume is one third of pi times the radius squared times the perpendicular height, written V = ⅓πr²h. A cone with a 4 cm radius and a 9 cm height holds 150.80 cubic centimetres.
Use Pythagoras on the radius and the perpendicular height: l = √(r² + h²). With r = 4 cm and h = 9 cm the slant height is √97, or 9.849 cm.
Lateral area is the curved sloping face only, πrl. Total surface area adds the flat circular base, giving πr(r + l). Use lateral area when the base is open, as on a funnel or a party hat.
Yes. Set the second measurement selector to slant height and the calculator recovers the perpendicular height as √(l² − r²). The slant height must be greater than the radius for a cone to exist.
A cone and a cylinder sharing the same base radius and height stand in an exact one to three ratio. That is why the volume formula is ⅓πr²h rather than πr²h, and it gives you a quick way to sanity check any answer.
For a point-down cone filled to half its height, only one eighth of the total volume. Partial volume scales with the cube of the fill ratio, so depth markings on a conical vessel are not proportional to contents.
The volume formula still applies if the height is measured perpendicular from the base plane to the apex, but the surface area formulas do not. An oblique cone has no simple closed form for its lateral area.
Results are mathematical values rounded for display. Real cones have wall thickness, seams and manufacturing tolerance, so allow a margin before cutting material or ordering to these figures.
