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CalcMax

Surface Area Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

131.95 cm²

Total surface area

A surface area calculator gives the area of the outside of a solid: how much material it would take to cover it, how much paint it would take to coat it, how much cardboard it would take to box it. Pick the shape and give it the measurements it reads — a radius for a sphere, a radius and a height for a cone or a cylinder, a length, a width and a height for a rectangular box — and the page returns one number in square centimetres. The four shapes here are the ones that cover almost every school and workshop question. A sphere is four π times the radius squared. A cone is π times the radius times the radius plus the slant height, where the slant height is the straight distance from the tip down to the rim and comes from Pythagoras on the radius and the height. A cylinder is two π times the radius times the radius plus the height, which is the two circular ends plus the curved side unrolled. A rectangular prism is twice the sum of the three pairs of faces: length times width, length times height, width times height. Four shapes, four formulas, and the reference tables below spell out which is which. The thing to know before you start is that all four length fields are on screen at once and stay filled in when you switch shape. That is a deliberate choice rather than an oversight — a form that hid the fields a shape does not use would have to move the boxes around under your hands as you changed your mind, and a form that kept them but ignored them would silently discard a number you had typed. So the page keeps every field where it is and reads only the ones the selected shape needs, and the rule for which those are is simple: a sphere reads the radius and nothing else, a cone and a cylinder read the radius and the height, and the rectangular prism reads the length, the width and the height. The radius field is not read by the prism and the length and width fields are not read by the other three, and the value sitting in a field that is not read changes nothing about the answer. The reference tables make this concrete, because their formula column says which row used which formula. One more thing worth knowing: the answer is in square centimetres regardless of the unit the dropdowns are set to, which is the batch's standing rule and applies to every page around this one.

Round solids reached from a radius and a height

FormulaRadius (cm)Height (cm)Surface area (cm²)
4πr²3—113.1
4πr²5—314.16
πr(r + l)3475.4
πr(r + l)512282.74
2πr(r + h)34131.95
2πr(r + h)512534.07
4πr²1—12.57
2πr(r + h)1218.85

Eight rows covering the sphere, the cone and the cylinder. The first column is the whole point of the table: once the shape is solved there is nothing left on the row to say which solid it was, so the formula stands in for the shape name. It also does something a shape name could not, which is tell you which of the two other numbers on the row was actually used — the two sphere rows have a dash in the height column because a sphere has no height and the field is not read at all. The first two rows are the sphere at radii of 3 and 5, and the ratio between them is worth checking: five thirds squared is about 2.78, and 113.1 times 2.78 is 314.16, which is the second row. Area goes with the square of the radius, never with the radius. The third row is the 3-4-5 cone whose slant height comes out exactly 5, and the fifth row is the cylinder page's own default: radius 3, height 4, surface area 131.95, the same figure that page prints as its total. The seventh row is a sphere of radius 1 at 12.57 square centimetres, which is 4π and is the only figure on the table that is a familiar constant rather than a measurement. The eighth is a cylinder of radius 1 and height 2 at 18.85, which is 6π. Every figure here is recomputed from its radius and height when the page is built, in centimetres, and the dashes are printed as they stand.

Rectangular boxes reached from a length, a width and a height

Length (cm)Width (cm)Height (cm)Surface area (cm²)
544112
12322
22224
101010600
0.50.50.51.5
34024

Six boxes, and the first row is the one the page loads with when the rectangular prism is selected. The second row is the smallest whole-number box, one by two by three, at 22 square centimetres — the arithmetic there is short enough to do in your head, which makes it the row to check the formula against: 2 × (2 + 3 + 6) = 22. The third row is a cube two centimetres on a side at 24, and the sixth row is a flat sheet three by four with a height of zero, also at 24. The pair is the most useful thing on this table, because the two solids look nothing alike and share a number: the cube has six faces of four, and the sheet has two faces of twelve and four that have collapsed. A surface area on its own does not pin down a shape, and this is the cheapest demonstration of that. The fourth row is a cube ten centimetres on a side at 600 square centimetres, which is the size of box a ream of paper arrives in, and the fifth is half a centimetre on a side, where the decimals start to matter. Every figure is exact, since the prism formula has no π in it, and all six are recomputed when the page is built.

Formula

Sphere: S = 4πr² Cone: S = πr(r + l), l = √(r² + h²) Cylinder: S = 2πr(r + h) Rectangular prism: S = 2(lw + lh + wh)

Shape
Which solid you are covering. It decides which of the four formulas runs, and therefore which of the length fields below are read at all — a sphere uses the radius alone, while the rectangular prism uses the length, the width and the height and ignores the radius entirely
r
The radius, in centimetres: the distance from the centre of the sphere to its surface, or from the centre of the circular face to its edge for a cone or a cylinder. Read by three of the four shapes and ignored by the rectangular prism
h
The height, in centimetres: the perpendicular distance from the base to the opposite face for a rectangular prism, or from the base up to the tip for a cone, or between the two circular ends for a cylinder. Ignored by the sphere, which has no height
l
Two different things, which is the one place this page asks you to read carefully. In the cone formula it is the slant height, the straight distance from the tip down the side to the rim, worked out from the radius and the height by Pythagoras. In the prism formula it is the length of the box. The formula column of the reference table says which row is which
lw, w, and the other letters
The rectangular prism's three dimensions in centimetres: length, width and height. The prism has six faces in three pairs, and each pair is one of the three products, which is why the formula doubles their sum rather than adding six separate terms
π
The ratio of a circle's circumference to its diameter, about 3.14159. It appears in three of the four formulas and not in the prism's, which is the reason a box's surface area is often an exact number while a sphere's almost never is
Square centimetres
The unit of the answer, always, whichever unit the dropdowns are set to. Surface area is a two-dimensional quantity, so the unit is the input unit squared — a radius in centimetres gives an area in square centimetres, and there is no cubic step anywhere on this page
Two decimal places
How wide the answer is written. Two rather than four because these are measurements of objects rather than coordinates on a plane: a radius taken off a ruler is already uncertain by more than the third decimal place

Most of the time the question is about material, and the shape tells you which formula you want. A ball, a globe or a balloon is a sphere and the question is usually how much fabric or paint covers it. A party hat, a funnel or a conical pile of sand is a cone, and the slant height is the measurement people forget — it is not the height of the cone but the distance down its side, and a cone of radius 3 and height 4 has a slant of exactly 5, which is Pythagoras in its smallest whole-number form. A tin, a pipe or a pillar is a cylinder, and the formula there is the one you can derive rather than memorise: two ends at πr² each plus a side that unrolls into a rectangle 2πr wide and h tall. A box, a crate or a room is a rectangular prism, and the six faces in three pairs is the whole idea. One warning carries across all four: surface area and volume are not proportional, and doubling every length multiplies the area by four and the volume by eight, which is why a large animal loses heat faster than a small one and why a small box costs disproportionately more cardboard than its contents suggest.

Worked examples

  1. A cylinder of radius 3 and height 4

    1. Two ends: 2 × π × 3 × 3 = 56.548667…
    2. Curved side unrolled: 2 × π × 3 × 4 = 75.398223…
    3. Total: 56.548667… + 75.398223… = 131.946891…, which rounds to 131.95
    4. The length and width fields are not read by the cylinder, and 5 and 4 sitting in them change nothing

    The shape the page loads with, and the row to compare against the cylinder page, which returns 131.95 as the total surface area for the same radius and height. Two pages, two tools, one figure — they run the same formula, and seeing them agree is the point. The check that needs no calculator: the two ends are about 28 each and the side unrolls into a rectangle about 18.8 wide and 4 tall, so the total should land a little above 130.

  2. A cone of radius 3 and height 4

    1. Slant height: √(3² + 4²) = √25 = 5
    2. Base: π × 3 × 3 = 28.274333…
    3. Curved side: π × 3 × 5 = 47.123889…
    4. Total: 28.274333… + 47.123889… = 75.398223…, which rounds to 75.4

    The 3-4-5 triangle doing its usual job: with a radius of 3 and a height of 4 the slant height is exactly 5, so the arithmetic stays clean all the way through. This is also the same 75.4 the cylinder page returns as a lateral area for a radius of 3 and a height of 4, which is a coincidence of two different solids rather than a shared formula — the cone's side and the cylinder's side are different shapes that happen to come to the same number here.

  3. A sphere of radius 3

    1. Four circles: 4 × π × 3 × 3 = 36π = 113.097335…
    2. Which rounds to 113.1
    3. Only the radius is read; the height, length and width fields are ignored by the sphere

    A sphere's surface area is exactly four times the area of its largest cross-section, which is the fact worth remembering about it — a circle of radius 3 is 28.27, and four of those is 113.1. The height field still holds 4 from the previous shape and the page takes no notice of it, which is the behaviour the intro describes and the reason it is described.

  4. A rectangular prism 5 by 4 by 4

    1. Top and bottom: 2 × 5 × 4 = 40
    2. Front and back: 2 × 5 × 4 = 40
    3. Two sides: 2 × 4 × 4 = 32
    4. Total: 40 + 40 + 32 = 112 square centimetres, which is exact

    The only one of the four shapes whose answer is routinely exact, and the only one that ignores the radius. Because the length and the height happen to be equal here, two of the three pairs of faces are the same size, which makes the sum easy to follow: 40, 40 and 32. The radius of 3 sitting in its field plays no part.

  5. A cube 2 by 2 by 2, and a prism 3 by 4 by 0

    1. Top and bottom: 2 × 3 × 4 = 24
    2. Front and back: 2 × 3 × 0 = 0
    3. Two sides: 2 × 4 × 0 = 0
    4. Total: 24 square centimetres

    The pair to look at together, because a cube two centimetres on a side and a flat sheet three by four both come to 24 square centimetres. The cube is 2 × 2 × 2 and gives 2 × (4 + 4 + 4) = 24; the second has a height of zero, so four of its six faces have collapsed to nothing and the two that remain are the top and the bottom. Same number, entirely different solids, and a reminder that a surface area on its own does not pin down a shape.

Limitations

The answer is always in square centimetres, whatever unit the dropdowns are set to. Choosing inches changes how your numbers are interpreted on the way in and nothing about how the answer is shown. All four length fields stay on screen for every shape, and the ones the selected shape does not read are ignored rather than cleared, so a radius left over from a sphere changes nothing about a box. The page will not stop you from leaving a field at a value that suits the previous shape, and it will not tell you that it did not use it. The formula column of the reference tables is the only place on the page that says which numbers went into a row, which is why it is there. Two decimal places is a display width rather than a claim about precision. The cone's surface area uses the slant height and not the perpendicular height, and the page computes the slant itself from the radius and the height, so a cone measured along its sloping side cannot be entered as given. The pages for a cylinder and for a cone elsewhere on this site return figures for some of the same inputs, and they agree with this page because they run the same formulas — a disagreement would be a bug rather than a difference of opinion. Nothing here handles a shape that is not one of the four, an open container whose ends are missing, or a solid with a hole through it.

Frequently asked questions

Which fields does each shape actually use?
The sphere reads the radius and nothing else. The cone and the cylinder read the radius and the height. The rectangular prism reads the length, the width and the height, and ignores the radius. Every field stays on screen and keeps whatever is in it, so a number left over from a shape you looked at earlier does nothing. The formula column of the reference tables is the quickest way to see which numbers went into a given row.
What is the slant height and how do I measure it?
It is the straight distance from the tip of the cone down its side to the rim, and it is longer than the vertical height unless the cone is flat. You do not have to measure it: the page works it out from the radius and the height using Pythagoras, since the radius, the height and the slant form a right triangle. A cone of radius 3 and height 4 has a slant of exactly 5.
Why is the answer in square centimetres when I typed in inches?
Because the answer is a fixed output in square centimetres, chosen so the results panel and the reference tables below it always agree. The dropdowns change how your measurements are understood on the way in, not how the answer is written out. To convert, divide the result by 6.4516 for square inches — or square the conversion factor, since area scales with the square of length.
Why does a cube 2 cm on a side have the same surface area as a 3 by 4 sheet?
Because both come to 24 square centimetres, and there is nothing more to it than that: the cube has six faces of 4 each, and the flat sheet has two faces of 12 and four that have collapsed to nothing because its height is zero. A single surface area does not identify a shape, and the second solid is a degenerate case the page is happy to compute rather than an error.
Do the cylinder and cone pages give the same answers as this one?
For the overlapping figures, yes, and that is by design rather than by accident — the three pages run the same formulas. A cylinder of radius 3 and height 4 has a total surface area of 131.95 here, and the cylinder page prints the same 131.95 as its total surface area, because both add the same two ends to the same unrolled side.
Can I work out how much paint or wrapping I need?
Yes, for a closed solid, and that is the commonest use of the page. Two things to watch: the answer is the whole outside, so an open container needs one face subtracted, and the figure takes no account of overlap, waste or thickness, which is why a practical estimate should be rounded up rather than used as it stands. The area is in square centimetres, so divide by ten thousand for square metres.

References

Related calculators

Cylinder Volume CalculatorVolume of a cylinder, plus the areas of its ends and its curved side.Cone Volume CalculatorVolume, slant height and surface area of a cone.Sphere CalculatorA ball: any one of radius, diameter or circumference gives volume and area.Cuboid Volume CalculatorVolume and surface area of a rectangular box.