Distance Calculator

What Is Distance?

Wolfram MathWorld puts it in thirteen words: the distance between two points is the length of the path connecting them. Everything difficult about the subject hides in the words "the path", because which path counts as the one depends entirely on the surface you are measuring across.

Distance on a Coordinate Grid

On a plane the shortest path is a straight line, and the straight line between two points is the hypotenuse of a right triangle whose legs run parallel to the axes. The horizontal leg is Δx = x₂ − x₁, the vertical leg is Δy = y₂ − y₁, and squaring both, adding, and taking the root gives the length. That is why entering (1, 2) and (4, 6) returns 5: the legs are 3 and 4, and the tool shows its work as √(9 + 16).

Two properties fall straight out of the squaring. Order never matters — (1,2) to (4,6) and (4,6) to (1,2) both return 5, because a negative difference squares to the same positive number. And negatives in the coordinates themselves are ordinary rather than special: (−3, −2) to (5, 4) returns 10, from Δx = 8 and Δy = 6, since subtracting a negative widens the gap rather than shrinking it.

The answer has no unit attached, and cannot have one. If your axes were feet, the 5 is 5 feet; if they were pixels, it is 5 pixels. The grid supplies the meaning and the arithmetic never sees it.

Distance on a Globe: The Great Circle

A sphere has no straight lines on its surface, so the shortest available path is the arc of a great circle — the circle you get by slicing the sphere through its center, which MathWorld names the orthodrome. Any other route between the same two points is longer.

This is why long-haul flight paths look bent on a wall map. The map is flat and the route is not; a New York to Tokyo track arcing up over the Arctic is the straight one, and the apparently direct line across the middle of the page is the detour. Map mode calculates that arc with the haversine formula on a sphere of radius 6,371 km, close to the 6,371.0084 km mean radius NASA's Jet Propulsion Laboratory publishes for Earth.

The scale of what a great circle can span shows up at the extremes. Entering (0, 0) and (0, 180) — two opposite points on the equator — returns 20,015.1 km (12,436.8 miles), which is half the planet's circumference and the longest answer the tool can produce. Equator to North Pole, (0, 0) to (90, 0), returns exactly half of that: 10,007.5 km (6,218.4 miles).

Distance, Displacement and Route Length

Three quantities get called distance in everyday speech, and only the first is what this page computes.

• Geometric distance — the length of the shortest path. This is the calculator's output: 5 on the grid, 3,935.7 km from New York to Los Angeles.
• Displacement — the same length carried together with a direction, which makes it a vector. The tool reports the magnitude but not the bearing, though Δx and Δy in coordinate mode contain everything you need to work the direction out.
• Route length — how far you actually travel. A road, a footpath or a shipping lane is never the arc, so this is always the largest of the three.

Δx and Δy do double duty. Divide the second by the first instead of squaring them and you get the gradient of the line joining the same two points, which is what you will find in the Slope Calculator.

One more distinction worth holding: distance is never negative. A negative Δy simply means the second point sits below the first, and the squaring erases the sign before the root is taken. (−2, 4) to (2, −4) returns 8.9443 from Δx = 4 and Δy = −8.

How Do You Calculate Distance?

One subtraction per axis, then a squaring, an addition and a square root. The map version adds trigonometry to bend the plane into a sphere, but the shape of the calculation is unchanged.

The Distance Formula, Written Out

The grid version is the one people mean by "the distance formula". The map version is the haversine, named for the half-versed-sine function at its center, and it is written in the form below because that arrangement stays numerically stable even when the two points are almost touching.

COORDINATE MODE
d = √( (x₂ − x₁)² + (y₂ − y₁)² )

MAP MODE (haversine)
a = sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2)
d = 2R · atan2( √a, √(1 − a) )

where φ is latitude in radians, λ is longitude in radians,
and R = 6,371 km

The two are closer relatives than they look. Both measure a separation by combining an east–west component with a north–south one; the haversine simply has to shrink the east–west component as you move away from the equator, which is the job the cos φ₁ · cos φ₂ product does. Set both latitudes to zero and that product becomes 1, leaving a formula that behaves like the flat one.

Coordinate mode is one theorem in disguise. If you would rather work with the two legs of the triangle directly instead of with four coordinates, the same arithmetic is waiting in the Pythagorean Theorem Calculator.

The miles figure is a straight unit conversion of the kilometer figure, using the factor 0.621371. That number is not an approximation of a fuzzy quantity — the international mile is defined as exactly 1,609.344 meters in NIST's conversion tables, and 1 ÷ 1.609344 is 0.621371 to six places.

Step by Step With This Calculator

The four input boxes are labeled generically — First, Second, Third and Fourth Value — because they change meaning with the mode. The placeholder text inside each one tells you which is which.

Order the points however you like. Latitude always comes before longitude within a point, but which point you call first makes no difference to the answer: New York to Los Angeles and Los Angeles to New York both return 3,935.7 km (2,445.6 miles).

• Set Point Type. "2D points (x, y)" is the grid; "Map points (lat, long)" is the globe.
• First Value: x₁ in grid mode, latitude of point 1 in map mode.
• Second Value: y₁ in grid mode, longitude of point 1 in map mode.
• Third Value: x₂ in grid mode, latitude of point 2 in map mode.
• Fourth Value: y₂ in grid mode, longitude of point 2 in map mode.
• Read the answer. The page loads pre-filled with 1, 2, 4 and 6 and already showing 5, so you can see the format before typing anything of your own.

Map coordinates must be decimal degrees, not degrees-minutes-seconds. Convert with degrees + minutes ÷ 60 + seconds ÷ 3600, and make the result negative for south or west. New York at 40°42′46″ N, 74°00′21.6″ W becomes 40.7128 and −74.0060, which the calculator measures to Los Angeles as 3,935.7 km (2,445.6 miles). A value typed as 40°42′46″, or as 40.7128N, is not a number and map mode answers "Enter both points' latitude and longitude (decimal degrees)" rather than guessing.

Worked Example: (1, 2) to (4, 6)

Grid mode, First Value 1, Second Value 2, Third Value 4, Fourth Value 6.

Δx = x₂ − x₁      4 − 1 = 3
Δy = y₂ − y₁      6 − 2 = 4
Square and add    3² + 4² = 9 + 16 = 25
Square root       √25 = 5
Midpoint          ((1+4)/2, (2+6)/2) = (2.5, 4)

What the result panel returns

5 · √((4 − 1)² + (6 − 2)²) = √(9 + 16) · Δx = 3, Δy = 4 — the Pythagorean theorem on the coordinate grid · Midpoint of the segment: (2.5, 4)

This is the 3-4-5 triangle, the oldest worked example in the subject and still the one tile-setters and framers use to square a corner. The midpoint is checkable against the tool itself: (1, 2) to (2.5, 4) returns 2.5, and (2.5, 4) to (4, 6) returns 2.5 as well, which is what a midpoint has to do.

Decimals behave identically. Entering 1.5, −2.25, 4.75 and 3.5 gives Δx = 3.25 and Δy = 5.75, shown as √(10.5625 + 33.0625), for a distance of 6.6049 and a midpoint of (3.125, 0.625).

Worked Example: New York to Los Angeles

Switch Point Type to "Map points (lat, long)" and enter New York at 40.7128 and −74.0060, then Los Angeles at 34.0522 and −118.2437. The two minus signs are the western longitudes and they are not optional.

What the result panel returns

3935.7 km (2445.6 miles) · Great-circle (haversine) distance between (40.7128, -74.006) and (34.0522, -118.2437) · As the crow flies — roads add 20–40% typically · Earth radius 6,371 km; accurate to ~0.5% (the Earth isn't a perfect sphere)

Round the same coordinates to two decimal places — 40.71, −74.01, 34.05 and −118.24 — and the answer moves to 3,935.2 km (2,445.2 miles). Half a kilometer of change across a 3,900 km measurement is a useful reminder that two decimal places already locate a city precisely enough for any travel-planning purpose.

Both figures carry the tool's own stated tolerance of about 0.5 percent, which on this route is roughly 19.7 km. Quoting 3,935.7 km to the nearest hundred meters is therefore a display convention rather than a claim about the tenth of a kilometer.

Distance Examples, Worked Out

Every figure in this section came out of the calculator at the top of the page, with the exact inputs shown alongside it.

Coordinate Examples

Grid mode. The Δ column is what the tool prints on its third line, and the midpoint is what it prints on its fourth.

The two middle rows are the degenerate cases and they are worth knowing. When Δy is zero the formula collapses to the plain horizontal gap, and when Δx is zero it collapses to the vertical one — √(36 + 0) and √(0 + 36) both return 6. Points that sit on the same row or column need no square root at all, which is a fast way to sanity-check a result you were not expecting.

Coordinate Pairs That Come Out Whole

Most coordinate pairs produce an irrational distance that the calculator rounds to four decimal places. A short list of pairs does not, because their legs form Pythagorean triples, and those make excellent test cases when you want to know whether you have entered something correctly.

Everything else lands on a decimal. The unit diagonal (0, 0) to (1, 1) returns 1.4142, which is √2 cut off at four places — the number that famously has no exact decimal at all, and the reason the calculator rounds rather than pretending.

City-to-City Examples

Map mode, with each city entered as its latitude then its longitude in decimal degrees. Every one of these is a great-circle arc, not a route.

London to Sydney at 16,993.9 km is 85 percent of the 20,015.1 km maximum any two points on the sphere can be apart, and it is the longest row here. That ceiling is a property of the geometry rather than of the cities: no pair of coordinates, however far apart they look on a flat map, can return more.

Examples That Have Nothing to Do With Maps

Coordinate mode does not care that its axes are usually called x and y. Anything you can place on a two-axis plan is fair input, and the diagonal is often the number you actually want.

• A room 12 ft by 16 ft: enter 0, 0, 12, 16 for a corner-to-corner diagonal of 20 ft — useful for checking whether a rug, a beam or a sheet of plywood will turn in the space.
• A 10 ft by 14 ft patio: 0, 0, 10, 14 returns 17.2047 ft, so the two diagonals of a correctly squared frame should both measure about 17 ft 2½ in.
• A screen or an image: entering its width and height as the second point returns the diagonal that display sizes are quoted in.
• A game board or a CAD drawing: any two cell references become two points, and the distance is the straight-line separation in cells or drawing units.

The construction use is the oldest one on the list. Because 3-4-5 is exact, a builder can mark 3 units along one edge and 4 along the other and know the corner is square when the diagonal measures precisely 5 — no protractor involved.

Distance Chart: Degrees, Kilometers and Miles

Latitude and longitude are angles, not lengths, and the ground distance one degree buys you depends on where you are standing. These charts convert between the two, and every figure was produced by running map mode on the relevant pair of points.

What One Degree Covers, by Latitude

Latitude lines are evenly spaced from equator to pole, so a degree of latitude is worth the same everywhere. Longitude lines converge as they climb, so a degree of longitude shrinks with the cosine of the latitude until it reaches nothing at all at the pole.

The 38° row is a deliberate check against an outside authority. The US Geological Survey uses 38° N — the latitude of Stockton, California and Charlottesville, Virginia — as its reference for exactly this question, and gives one degree of latitude as about 69 miles and one degree of longitude as about 54.6 miles. This calculator returns 69.1 and 54.4 miles for the same pair of measurements, agreeing to a fraction of a percent, with the small residual coming from the sphere the haversine assumes against the flattened ellipsoid the USGS figures rest on.

The left-hand column also explains where the number 111.2 comes from. A sphere of radius 6,371 km measures 2π × 6,371 = 40,030.2 km around, and dividing that by the 360 degrees in a full circle gives 111.195 km per degree, which map mode prints as 111.2. Working the other way, do not multiply the rounded 111.2 back up: 111.2 × 360 gives 40,032 km, nearly 2 km past the model's real circumference. Half that circumference, 20,015.1 km, is the calculator's own longest-possible measurement — the figure it returns for two opposite points.

Decimal Places and Precision on the Ground

How many decimal places a coordinate needs is a question about how precisely you are trying to point. Measured from 40° N along a meridian, each place you drop costs you a factor of ten.

Four decimal places is where map mode runs out of resolution, because kilometers are printed to one decimal. That last row is a real distance of about 11 meters being displayed as 0.0 km. The tool is built for city-to-city work and reports at that scale.

Two places is enough for any journey planning. Three is enough to identify a building. Past four, use a surveying tool rather than this one.

Kilometers and Miles

Map mode prints both units on the same line, so no conversion is needed — but the conversion is worth knowing, because the factor is exact rather than approximate. NIST's conversion tables define the international mile as exactly 1,609.344 meters, and 1 ÷ 1.609344 gives 0.621371, which is the multiplier this calculator applies.

Going the other way, one mile is 1.6093 km. A quick mental version that is wrong by about 3 percent: halve the kilometers, then add a tenth of the original — 100 km becomes 50 + 10 = 60, against the true 62.14.

How to Read Your Result

The panel returns a headline number and three supporting lines. There are no bands or categories to interpret, so the reading is mostly a matter of knowing what each line is for and which of them can quietly mislead you.

The Four Lines in Coordinate Mode

Running (1, 2) to (4, 6) produces these, in this order:

• The distance itself, rounded to four decimal places: 5. Trailing zeros are dropped, so a whole answer prints as a whole number.
• The substitution, with your own numbers in it: √((4 − 1)² + (6 − 2)²) = √(9 + 16). This is the line to check first if an answer looks wrong, because it shows exactly which values the tool read from the boxes.
• The two differences: Δx = 3, Δy = 4. Their signs tell you the direction — a negative Δy means the second point is below the first.
• The midpoint: (2.5, 4).

Four decimal places is generous for most work and still finite. Coordinate distances below 0.00005 units display as 0, so (0, 0) to (0.0001, 0) returns 0.0001 while (0, 0) to (0.00001, 0) returns 0. If your coordinates are that tightly packed, rescale them by a thousand before entering and divide the answer back down.

The Four Lines in Map Mode

Map mode leads with both units on one line — 3,935.7 km (2,445.6 miles) — then restates the two coordinate pairs it used, then carries two standing caveats: that the figure is as the crow flies with roads typically adding 20 to 40 percent, and that it assumes a 6,371 km sphere and is accurate to about 0.5 percent.

The restated-coordinates line is the one to actually read. It echoes the four numbers back as (latitude, longitude) pairs, which is the fastest way to catch a value that landed in the wrong box or lost a minus sign in transit.

Trailing zeros are stripped from that echo, so entering −74.0060 comes back as -74.006. The value is unchanged; only the display is tidier.

The Midpoint Line, and Where It Stops Working

In coordinate mode the midpoint is the plain average of each axis, ((x₁+x₂)/2, (y₁+y₂)/2), and it is genuinely the halfway point. For (1, 2) to (4, 6) the midpoint is (2.5, 4), and measuring each leg separately confirms it: (1, 2) to (2.5, 4) returns 2.5, and (2.5, 4) to (4, 6) returns 2.5.

That averaging trick does not survive the move to a sphere, which is why map mode does not report a midpoint. Averaging New York and Los Angeles gives (37.3825, −96.1249); measuring the two legs through it returns 1,940.2 km and 2,026.0 km. They are not equal, and they add to 3,966.2 km against the direct 3,935.7 km — a 30.5 km detour introduced purely by treating the globe as flat.

A true great-circle midpoint needs spherical interpolation rather than an average. Over short distances the error is negligible; across an ocean it puts your halfway marker in the wrong place by tens of kilometers.

When the Answer Is Zero, or Just Surprising

An empty box is read as zero rather than rejected, and that is the single most likely cause of a result you did not expect. Leaving First Value blank while the others hold 2, 4 and 6 does not raise an error — it computes (0, 2) to (4, 6) and returns 5.6569. In map mode a blank latitude quietly places the point on the equator.

A genuine zero is also possible: identical points return 0, as (3, 3) to (3, 3) does. Text the tool cannot read is the one input it refuses outright, replacing the readout with "Enter both points' coordinates". Note that a number typed with a thousands separator, like 1,000, counts as text and triggers that message — strip the comma.

Map mode adds one range guard. A latitude beyond ±90 or a longitude beyond ±180 returns "Latitude ±90, longitude ±180" instead of an answer, which catches many, though not all, longitude-first entry mistakes.

Straight-Line Distance vs Road Distance

Every map answer on this page is the arc a bird would fly. No road, railway or shipping lane follows it, so the number you get here is always shorter than the journey — the question is only by how much.

The 20 to 40 Percent Band

The allowance this calculator states is 20 to 40 percent on top of the great-circle figure. Applied to four map-mode results, that produces the following envelopes.

Treat the band as a planning envelope, not a prediction. It is a rule of thumb about how road networks behave, and the only way to know a specific route is to ask a routing service that has the actual road graph.

What Pushes a Route to the Top or Bottom of the Band

The gap between the arc and the road is a function of what the road had to avoid, and the pattern is fairly consistent.

• Long routes across open, gridded terrain sit at the bottom of the band or below it. A transcontinental interstate can run close to straight for hundreds of miles at a stretch, so the percentage added is small.
• Short urban trips sit at the top. Over a few kilometers a one-way system, a river or a railway line can add a large fraction to a small number, and the arc becomes a poor predictor of anything.
• Water and mountains dominate everything else. Any journey that has to reach a bridge, a tunnel or a pass is set by where that crossing is, not by where the two endpoints are.
• Islands and inlets can break the band entirely. Two points a few kilometers apart across a bay may have no road connection shorter than a long detour inland.

The direction of the error is the useful part: the arc is a hard floor. Nothing on the ground can be shorter than the great circle, so this number is a genuine lower bound on any journey, and a fuel, time or cost estimate built on it will always be an underestimate rather than an overestimate.

Aviation and shipping are the exceptions that prove it. Aircraft and vessels can follow something close to the arc, which is why a flight distance usually sits within a few percent of the figure here, adjusted for winds and airspace rather than for roads.

Limits: When This Calculator Does Not Apply

The tool measures the shortest geometric path between two points on a plane or on a sphere. Being precise about the edges of that job is more useful than a longer list of things it does not do.

• It does not do three dimensions. There is no z field. The formula extends by adding (z₂ − z₁)² under the root, so a 300-unit ground run with a 40-unit rise is √(300² + 40²) = 302.6549 — run the flat part here and finish that step yourself.
• It cannot tell latitude from longitude. Entering London and Paris longitude-first returns 403.6 km instead of the correct 343.6 km, with no warning, because every value involved is inside the legal range. The ±90 and ±180 guard only catches a swap when the longitude happens to exceed 90, as it does for New York and Los Angeles.
• It treats Earth as a sphere. NASA's JPL publishes an equatorial radius of 6,378.1366 km against a mean of 6,371.0084 km — the equator bulges about 7.1 km beyond the average the haversine uses. That is the source of the stated 0.5 percent tolerance, and it is why survey and legal boundary work uses ellipsoid geodesics instead.
• It gives no bearing. You get the length of the arc but not the direction of travel, and on a great circle the bearing changes continuously along the route anyway.
• It assumes decimal degrees. Degrees-minutes-seconds, and coordinates carrying N/S/E/W letters, are text and will be rejected rather than converted.
• An empty box counts as zero rather than raising an error, so any answer that arrives without a complaint is still worth checking against the echoed coordinates on the second line.
• It measures point to point only. There is no waypoint list and no running total, so a multi-leg journey has to be run one leg at a time and added up by hand.

Two things that look like limits are not. Coordinate mode handles very large values without trouble — entering 0, 0, 1000000 and 1000000, with no separators, returns 1,414,213.5624 — and map mode crosses the antimeridian correctly: 179° E to 179° W returns 222.4 km (138.2 miles), the same answer as any other two-degree separation at the equator, rather than the 358 degrees the wrong way around.