Convert cm to km

Centimetres are the everyday metric length — body measurements, paper, furniture — and they are the one metric unit that has no direct customary counterpart, which is why so much converting happens through them.

Kilometres are the road distance nearly everywhere outside the United States and the United Kingdom, which is why converting them is usually about reading a map or a race entry.

Going from centimetres to kilometres moves the decimal point 5 places to the left and changes nothing else. There is no factor to remember and no rounding to decide: the digits stay in the same order and only their position changes.

1,234 cm is 0.01234 km — same digits, moved along. That is worth knowing because it is the one kind of conversion you can check at a glance: if the digits of the answer are not the digits you started with, something other than the conversion has happened to them.

centi- and kilo- are not part of the unit — they are a multiplier bolted onto the front of it, and the same one means the same thing on every unit it appears on.

Here that is a hundredth of a metre against a thousand metres: 5 powers of ten apart, and the unit underneath both of them is the same metre. Nothing about the quantity changes — only how many of it you are counting in one go.

A hundred thousand centimetres make a kilometre, and that is the number every metric map scale is built around. A 1:100,000 map puts one centimetre of paper on one kilometre of ground, so the ratio and the conversion are the same statement. Every other scale is a fraction of it: 1:50,000 gives two centimetres to the kilometre, 1:25,000 gives four, 1:10,000 gives ten.

Which means the arithmetic on a map is simpler than it looks. Measure the centimetres, divide by the number of centimetres in a kilometre at that scale, and you have kilometres. The only figure to carry is that one small number — two, four or ten — and it is printed on the sheet in the legend beside the ratio.

Topographic maps at 1:25,000 and 1:50,000 carry a grid whose squares are one kilometre on a side, and counting squares is faster and more reliable than laying a ruler along a route. A leg crossing three squares diagonally is roughly four kilometres, because the diagonal of a one-kilometre square is 1.41 km, and that figure is worth remembering because so many walked legs run at an angle to the grid.

The grid also gives a free check on any ruler measurement. Measure one grid square with the same ruler you are using for the route and see whether it comes out at the four or two centimetres the scale claims. If it does not, the sheet has been resized somewhere between the survey and your hands, and every distance you take off it will be wrong in the same proportion.

The printed ratio describes the map as it left the cartographer. The scale bar describes the piece of paper in front of you, and those two stop agreeing the moment anything is copied, printed to fit, or exported at a different page size. A map reduced to ninety per cent still says 1:25,000 in the legend and is now 1:27,778, which puts every kilometre out by a tenth.

Screens are worse, because a browser has no idea how large your display is. A map that renders at a stated scale on one monitor renders at another on a phone, and a ruler held against the glass measures the display rather than the map. Digital maps handle this by drawing a bar that redraws itself at every zoom level, which is an admission that the ratio alone means nothing on screen.

A straight ruler measures the shortest line between two points and a path almost never takes it. The traditional fixes both work: lay a piece of string along the route and then measure the string, or run a map wheel along it and read the dial. Either one recovers what a ruler cuts off at every bend, and on a winding valley path that can be a fifth of the total.

Breaking the route into short straight legs and adding them is the compromise most people actually use, and its accuracy is entirely a function of how short the legs are. Legs of a couple of centimetres pick up most of the shape; legs the length of the ruler pick up almost none of it. Measure the same route twice with different leg lengths and the difference between the answers is the error you are carrying.

A map is a projection onto a horizontal plane, so a slope is drawn as its horizontal extent and the climb is recorded only by the contours crossing it. A kilometre measured across a steep face is a longer kilometre on the ground, though the difference is smaller than intuition suggests: even a one-in-five gradient adds only about two per cent to the distance travelled.

The effect on time is much larger than the effect on distance, which is what Naismith's rule from 1892 was written to capture. In its metric form it allows about an hour for every five kilometres of ground covered plus a minute for every ten metres of ascent, so a five-kilometre leg with three hundred metres of climb takes about an hour and a half rather than an hour. The distance was never the thing to plan around.

Every map has a resolution limit set by the width of the lines drawn on it. A line half a millimetre across — a fine pencil, a printed path — is 0.05 cm, which stands for 12.5 m of ground at 1:25,000 and 25 m at 1:50,000. Nothing narrower than that can be represented, and nothing measured off the sheet can be more precise than that.

So a route measured on paper is honest to about a tenth of a kilometre and no better, whatever the ruler suggests. Quoting a walk as 12.4 km from a map measurement is fine; quoting it as 12.43 km is claiming a resolution the paper does not have. The right response to a figure with too many digits is to round it, not to measure again more carefully.

Alongside the grid and the scale bar, a map carries latitude and longitude, and those give a distance measure that needs no scale at all. A minute of latitude is one nautical mile, which is 1,852 m exactly by definition, so sixty minutes — one degree — is about 111 km. The graticule ticks along the edge of the sheet are therefore a second ruler, calibrated in the ground rather than in the paper.

It works only in one direction. Lines of longitude converge towards the poles, so a minute of longitude is a full nautical mile at the equator and progressively less further north or south — at the latitude of Britain it is a little over half. Measuring north to south against the graticule is reliable anywhere; measuring east to west against it is a mistake that grows with latitude.

Digital mapping removes the ruler and the ratio together. A measuring tool follows a drawn line along a projected surface and returns a distance directly, which eliminates the chord error and the photocopy error in one move, and it will happily report a figure to the metre. That precision is about the geometry of the drawn line, not about the route.

The remaining error is the one that never went away: what you drew is not what you will walk. A line snapped to a path is as good as the path data, a line drawn freehand is as good as your hand, and neither knows about a diversion, a closed gate or the width of a track. Paper and screen converge on the same advice — take the distance to a tenth of a kilometre and plan the time separately.