How this instrument works
The distance from a point (x,y) to the origin, (0,0), is found the same way as the distance between any two points — treating the point's own x and y coordinates as the two legs of a right triangle, with the origin sitting at the triangle's third corner, and applying the Pythagorean theorem to find the hypotenuse.
This is exactly the length, or magnitude, of the point's own position vector — the arrow pointing from the origin straight out to that point. Because one of the two points involved is always fixed at (0,0), the general two-point distance formula simplifies down to just the square root of x squared plus y squared, with no subtraction needed at all.
This page is a narrower, single-point version of this site's general two-point Euclidean Distance calculator, which takes two independent points rather than assuming one of them sits at the origin.
- Enter the point's x-coordinate into the x field.
- Enter its y-coordinate into the y field.
- Read Distance from the origin: the straight-line length from (0,0) to that point.
- Compare against the general Euclidean Distance calculator on this site by fixing one of its two points at (0,0) — the results should match exactly.
Worked example — the point (3,4)
The point (3,4) sits exactly √(3²+4²)=√25=5 units from the origin — a 3-4-5 right triangle formed by the point's own coordinates and the origin, with the point's distance from (0,0) as the hypotenuse.
The point (5,12) sits √(5²+12²)=√169=13 units from the origin — a 5-12-13 right triangle. The point (0,7) sits exactly 7 units from the origin, since its x-coordinate is 0, leaving the distance equal to just its y-coordinate's own magnitude, with no diagonal component involved at all.
Questions
What is the distance from a point to the origin?
d = √(x²+y²), found via the Pythagorean theorem using the point's own x and y coordinates as the two legs of a right triangle whose hypotenuse reaches from the origin out to that point.
Is this the same as the magnitude of a vector?
Yes — the distance from the origin to a point (x,y) is exactly the magnitude (length) of that point's own position vector, the arrow drawn from (0,0) straight out to it.
How is this different from the Euclidean Distance calculator on this site?
That page takes two independent points and finds the distance between them; this page is the narrower special case where one of those two points is always fixed at the origin, simplifying the formula to just x²+y² under the root.
What if x or y is negative?
It makes no difference to the result — both coordinates are squared before adding, and squaring a negative number gives the identical positive result as squaring its positive counterpart.
What is the distance from the origin to itself?
Exactly 0 — a point with coordinates (0,0) sits at zero distance from the origin, since it IS the origin, with no separation to measure at all.