Cone Volume Calculator

V = πr²h ÷ 3 l = √(r² + h²) Lateral area = πrl Total surface area = πr(l + r)

r

The radius of the circular base, in centimetres. Required, and zero is allowed: a cone of radius zero is a vertical segment standing on a point

h

The perpendicular height from the centre of the base straight up to the apex, in centimetres. Required, and it is not the slant height — the two are different numbers on every cone except the degenerate one where the height is zero

l

The slant height, the distance from the apex down the side to the rim. It is the hypotenuse of the triangle formed by the radius, the height and the side, so it is always longer than either of them. It is the only one of the four outputs that is routinely exact, since a whole-number radius and height can give a whole-number slant

V

The volume in cubic centimetres, one third of the base area times the height. The third is the whole difference between this page and the cylinder page, where the same expression has no divisor

Lateral area

The area of the curved surface alone, π times the radius times the slant height. It does not include the base. Note that it uses the slant height and not the height, which is the single most common substitution error on this shape

Total surface area

The curved surface plus the circular base, which simplifies to π times the radius times the slant height plus the radius. It is worked out from the radius and the slant height in one step rather than by adding the two areas, so it is not always the lateral area on screen plus πr²

Two decimal places

How wide every output is written. The volume and the two areas are irrational for almost every input; the slant height is exact whenever the radius and the height form a Pythagorean triple, which is why the page's default values produce trailing zeros

The obvious jobs are the ones where you have a cone in front of you and need a number about it. A conical pile of gravel or grain, where you can measure the width of the base and the height but the slant is buried inside the material. A conical roof or a spire, where the slanted surface is what has to be covered and the slant height is the number the roofing is priced on. A funnel, a hopper, a paper cup, a party hat — the volume tells you what it holds and the lateral area tells you how much material it takes to make, and those are two different questions with two different answers on the same page. The third question the page answers is the comparative one. Choosing between a cone and a cylinder is a real decision in containers and in storage, and the comparison is always the same factor of three: the cylinder of the same base and height holds exactly three times as much, so a conical hopper of a given footprint holds a third of what a cylindrical one does. Two habits are worth carrying. The slant height is always the longest of the three lengths in the triangle down the side, so a slant that comes out shorter than the height means the two boxes were filled the wrong way round — the slant is the one you would measure along the outside, and it cannot be the smaller. And the lateral area is always less than the total surface area, since the total is the lateral plus the base, so a total smaller than the lateral is a sign that something was subtracted rather than added.

1. A radius of three and a height of four

   1. Slant height: √(3² + 4²) = √25 = 5
   2. Volume: π × 3² × 4 ÷ 3 = π × 36 ÷ 3 = 12π = 37.6991…, which rounds to 37.70
   3. Lateral area: π × 3 × 5 = 15π = 47.1239…, which rounds to 47.12
   4. Total surface area: π × 3 × (5 + 3) = 24π = 75.3982…, which rounds to 75.40

   The input the page loads with, and the only one in this set where every step can be done mentally: three and four give a hypotenuse of exactly five, so the slant height is a whole number and the two areas are whole multiples of π. Note the volume arithmetic — the base area is nine π and a third of that times four is twelve π, which is why the answer is a little under thirty-eight rather than a little over a hundred. A cone of this size holds about thirty-eight millilitres, which is a useful thing to know when a result looks wrong by a factor of three.

2. A radius of five and a height of twelve

   1. Slant height: √(25 + 144) = √169 = 13
   2. Volume: π × 25 × 12 ÷ 3 = 100π = 314.1592…, which rounds to 314.16
   3. Lateral area: π × 5 × 13 = 65π = 204.2035…, which rounds to 204.20
   4. Total surface area: π × 5 × (13 + 5) = 90π = 282.7433…, which rounds to 282.74

   The other well-known whole-number triple, and a taller, narrower cone than the default. Notice how close the volume is to a round number: a hundred π is 314.16, so a cone of radius five and height twelve holds almost exactly a litre. That is the kind of coincidence worth using as a check when you are working in centimetres — a hundred π cubic centimetres is a hundred times the volume of a one-centimetre cube scaled by π, and the numbers on the screen should be a little over three hundred.

3. A radius of six and a height of eight, where two answers coincide

   1. Slant height: √(36 + 64) = √100 = 10
   2. Volume: π × 36 × 8 ÷ 3 = 96π = 301.5928…, which rounds to 301.59
   3. Lateral area: π × 6 × 10 = 60π = 188.4955…, which rounds to 188.50
   4. Total surface area: π × 6 × (10 + 6) = 96π = 301.5928…, which rounds to 301.59

   The row on the table below that gets reported as an error. The volume and the total surface area are the same number, 301.59, and that is arithmetic rather than a duplicated column: the volume works out as 96π and the total surface area also works out as 96π, because with a slant of ten the expression πr(l + r) becomes π × 6 × 16, which is the same ninety-six π as the volume fraction. The two are not the same kind of quantity — one is cubic centimetres and the other is square centimetres — so the coincidence is in the digits and nowhere else. It happens because this particular cone has a slant height equal to eight and a radius and height that form a triple; it is not a rule, and the neighbouring rows do not share it.

4. A radius equal to the height

   1. Slant height: √(100 + 100) = √200 = 14.1421…, which rounds to 14.14
   2. Volume: π × 100 × 10 ÷ 3 = 1000π ÷ 3 = 1047.1975…, which rounds to 1047.20
   3. Lateral area: π × 10 × 14.14214 = 444.2883…, which rounds to 444.29
   4. Total surface area: π × 10 × (14.14214 + 10) = 758.4479…, which rounds to 758.45

   A cone as tall as it is wide across the base radius, which makes the triangle isosceles and the slant height exactly the radius times the square root of two. That square root is irrational, so this is the ordinary case: a slant height that goes on forever and two areas that follow it. It is also the case where the volume formula is easiest to read — a thousand π over three — and where a reader who has forgotten the third will land on 3141.59 instead, which is exactly three times too much. The factor of three is the single most common error on this page.

5. A height of zero

   1. Slant height: √(16 + 0) = 4, which is the radius, because the apex has not been lifted at all
   2. Volume: π × 16 × 0 ÷ 3 = 0, since there is no height to give it any
   3. Lateral area: π × 4 × 4 = 16π = 50.2655…, which rounds to 50.27
   4. Total surface area: π × 4 × (4 + 4) = 32π = 100.5310…, which rounds to 100.53

   A legal input that produces a surprising row: the volume is zero and the other three answers are not. A cone of height zero is a flat disc — there is nothing above the base, so it holds nothing, but its curved surface has not disappeared, it has flattened out into a second disc the same size as the base, and the total area of the two faces is twice a single disc. This is worth seeing once because a zero volume with three non-zero answers beside it looks like a partially failed calculation and is not. It is also the input that catches the arithmetic shortcut described above: the total surface area here is πr² + πr², which is exactly the lateral area doubled, and a page that added the two would get the same answer on this row and a different one on the first row.

Every output is in centimetres, square centimetres and cubic centimetres, whatever the input dropdowns are set to. Choosing inches changes how your numbers are read on the way in and nothing about the results, so a cone entered as four inches by six inches comes back with a volume in cubic centimetres. This is deliberate and the reference table follows the same convention, but it means the answer needs converting if you wanted it in the unit you typed. The volume is reported in cubic centimetres and not in litres or millilitres, which are a different unit in the same dimension; if you need litres, divide by a thousand, and if you need millilitres the number of cubic centimetres is the number of millilitres. Three of the four answers contain π and are therefore irrational for almost every input, so two decimal places is a cut rather than a complete answer. The four answers are computed from your radius and height directly and never from one another, so the total surface area is not always the lateral area on screen plus the base area on screen — it is the same expression evaluated in one step, and reconstructing it from the printed lateral area can differ in the last digit. This page takes two measurements and gives four; it will not work backwards from a volume to a radius, and it has no box for the slant height, so a cone whose slant you can measure but whose height is buried in material cannot be entered here as it stands. The height is the perpendicular height and not the slant, and the page cannot tell that the two have been confused — feeding a slant height into the height box gives four answers that are all internally consistent and all wrong. At least one of the two inputs must be non-zero for a meaningful answer, though both inputs may be zero if you want the arithmetic rather than the shape. Finally, this is a right circular cone with its apex directly above the centre of the base; a cone leaning to one side, or one whose base is an ellipse, is a different solid and the formulas here do not apply to it.

Why is the volume divided by three?

Because a cone holds exactly one third of what a cylinder on the same base with the same height holds. Take the base area, multiply by the height, and you have the volume of that cylinder; the cone is one third of it. The same relation holds between a pyramid and the prism it would fit inside, and it is the single most commonly forgotten factor in this calculation — a result that is three times too large is almost always a missing division by three rather than an arithmetic slip.

Is the height the same as the slant height?

No, and this is the mistake this shape invites most often. The height is measured straight up from the centre of the base to the apex. The slant height is measured along the surface, from the apex down to the rim, so it is the hypotenuse of the triangle formed by the radius, the height and the side, and it is always longer than either of the other two. The volume uses the height. The lateral area uses the slant height. Confusing them produces an answer that is wrong by a factor that depends on how tall the cone is, and the page cannot detect it.

How do I work out the slant height?

It is the square root of the radius squared plus the height squared, which is the Pythagorean theorem applied to the triangle down the side of the cone. A radius of three and a height of four give a slant of exactly five, which is why those are the numbers the page loads with: every step is checkable in your head. Most pairs of numbers do not work out so tidily, and the slant height is then an irrational number rounded to two decimal places.

Why is the total surface area not just the lateral area plus the base?

In exact arithmetic it is, and the page uses the combined form πr(l + r) rather than adding two separately rounded pieces. The difference shows up in the printed digits: the lateral area is rounded to two places before you ever see it, so adding it to the base area yourself can land a hundredth away from the answer the page gives. The page's figure is the one computed from the radius and the slant height in a single step, and it is the one to trust.

Why is the answer in centimetres when I typed in inches?

Because this page reports everything in centimetres, square centimetres and cubic centimetres regardless of the dropdowns, so that the results and the reference table below them always agree. The dropdown changes how your number is understood on the way in. To convert, divide the length by 2.54, the areas by 6.4516 and the volume by 16.3871; or leave the conversion until the end and change the unit of the answer, which is usually less work.

Can the radius or the height be zero?

Yes, and both give real answers rather than empty ones. Setting the height to zero flattens the cone into a disc: the volume is zero because nothing is above the base, but the curved surface has simply become a second disc, so the lateral area equals the base area and the total surface area is twice it. Setting the radius to zero collapses the cone into a vertical segment: the volume and both areas are zero, while the slant height equals the height. Setting both to zero gives zero everywhere.

• Cone — the solid formed by a circular base and an apex, with the volume one third of the base area times the height and the lateral area πrl, both of which this page evaluates; the entry traces the one-third result to Eudoxus, with later proofs by Archimedes in On the Sphere and Cylinder and by Euclid in Elements XII.10 — Wolfram MathWorld (United States)
• Pythagorean theorem — the relation that gives the slant height from the radius and the perpendicular height, since those two are the legs of the right triangle down the side of the cone — Wolfram MathWorld (United States)
• Pi — the constant in the volume and the two areas, and the reason three of the four answers on this page are irrational for almost every input — Wolfram MathWorld (United States)
• 教育部关于印发义务教育课程方案和课程标准（2022 年版）的通知——The fifth item in the annex list of this notice is the Mathematics Curriculum Standards for Compulsory Education (2022 edition); the volume and lateral surface of a cone and its slant height are part of the compulsory-education mathematics curriculum, and the wording of the standards and the grade-band breakdown are governed by that annex — 中华人民共和国教育部