Parallax Distance Calculator

d (pc) = 1 / p (arcsec) — the trigonometric parallax method anchors the entire cosmic distance ladder.

Worked example — Parallax Distance Calculator

Distance (parsecs)	10 pc
Distance (light-years)	32.6156 ly
Distance (AU)	2062650 AU

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Parallax Distance Calculator

Parallax distance is measured by triangulation, with the Earth’s orbit as the baseline. Watch a nearby star in January and again in July and it appears to shift against the far background; half that total shift is the parallax angle p, and it is the angle the Earth–Sun distance subtends as seen from the star.

What each symbol means

p: the parallax angle in arcseconds — half the annual back-and-forth, not the whole of it. One arcsecond is 1/3600 of a degree, and no star has one: the largest real parallax on the sky is under 0.8″.
d: the distance, which is the reciprocal of the angle and nothing more. Halve the angle and you double the distance: 0.1″ gives 10 pc, 0.02″ gives 50 pc.
pc: the parsec, and it is not an independent unit of length so much as a bookkeeping choice — the distance at which the parallax comes out to exactly 1″. That definition is the whole reason the formula has no constant in it.
ly: the light-year, the same distance in the unit non-astronomers use. One parsec is 3.2616 of them, so the second row is always the first multiplied by that figure.

Where the formula comes from

The Earth–Sun distance 1 AU is the short side of a very thin right triangle, and the distance to the star is the long one. The angle at the star is p.
For a thin triangle tan p ≈ p when p is in radians, and the error in that swap is smaller than any parallax has ever been measured to. So d = 1 AU / p, with p in radians.
Arcseconds are more convenient than radians, and converting introduces a constant — the number of arcseconds in a radian, about 206265. So d = 206265 AU / p″.
Rather than carry that constant forever, astronomers named it: one parsec is 206265 AU. Divide through and the formula collapses to d = 1/p, which is the line the top row prints. Set the angle to 1 and the distance reads exactly 1 pc.

How to read what you see

The reciprocal is easiest to see by halving the angle rather than by reading one value: 0.77″ gives 1.2987 pc, 0.1″ gives 10 pc, 0.02″ gives 50 pc. The light-year row tracks the first one rigidly at 3.2616 to the parsec — 10 pc shows 32.6156 ly — so it carries no information the first row does not, only a change of unit.

Assumes: That the shift you measured is parallax and not the star’s own motion across the sky, and that you halved the annual swing rather than using all of it. Both mistakes scale the answer by a clean factor, which makes them hard to spot in a single measurement.
Breaks when: The formula never breaks; the measurement does, and it does so at a distance rather than at a value of the formula. Because d = 1/p, a fixed error in the angle becomes an error in distance that grows with the square of the distance — at 0.02″ an uncertainty of 0.002″ is 10% of the angle but moves the answer between about 45 and 55 pc. This is why parallax is the first rung of the distance ladder and not the whole of it.

The angle is so small it nearly sank heliocentrism 🖖

If Earth really moves, nearby stars must appear to shift over the year. Copernicus knew that in 1543, and so did his critics, who pointed out that no such shift had ever been seen and concluded the Earth stands still. They were reasoning correctly from the evidence — the angles are simply minute. Even Proxima Centauri, the closest star there is, has a parallax of only 0.7685 arcseconds, roughly the width of a coin seen from three kilometres. Nobody could measure that for nearly three centuries. Friedrich Bessel finally did it in 1838 with 61 Cygni, at about 0.31 arcseconds, and stellar parallax stopped being an objection to heliocentrism and became the first rung of the distance ladder.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

Insight 1 says Friedrich Bessel measured 61 Cygni at about 0.31 arcseconds in 1838, the first stellar distance anyone obtained. Type it in. How far away is it, and how does that compare with Proxima Centauri's 0.7685″?

Show answer

3.2258 pc, or 10.5212 ly. Proxima at 0.7685″ reads 1.3012 pc and 4.2441 ly, so Bessel's star is about two and a half times further away. He was not aiming at the nearest star — he picked 61 Cygni because it moves quickly across the sky, which is a different clue and a good one: a star with large proper motion is usually a star that is close.
The angle box stops at 2 arcseconds. Set it there. What distance comes back, and what is wrong with that picture?

Show answer

0.5 pc, which is 1.6308 ly — and nothing is there. The nearest star to the Sun is Proxima Centauri at 4.2441 ly, so everything above 0.7685″, the whole top half of this input, names a region of space with no star in it. The formula would go further still: d = 1/p has no smallest distance. The box stops before the answer becomes absurd, not because the arithmetic does.

Problem solved in full

Distance of a star that shifts by 0.1 arcseconds as Earth moves 5 steps
A star shifts by 0.1 arcseconds as Earth moves across its orbit. Find its distance — then find the distance at which this method stops working, and why.
1. Parallax is a triangle with a known base: Earth’s orbital radius. Over six months the baseline is 2 AU, and half the total shift is the parallax angle p. At these angles the tangent is the angle itself in radians, which is what makes the arithmetic trivial.
  
  $tan p = \frac{1 AU}{d} \approx p (radians)$
2. The parsec is defined to make it trivial: it is the distance at which 1 AU subtends one arcsecond. Choose that unit and the relation is a reciprocal with no constants at all.
  
  $d [pc] = \frac{1}{p [ arcsec ]} = \frac{1}{0.1} = 10 pc$
3. Converting is just unit bookkeeping, and the calculator above prints all three of these for p = 0.1″.
  
  $10 pc \times 3.26156 = 32.6156 ly, 10 \times 206265 = 2062650 AU$
4. Now the part the tool does not show. Because the relation is a reciprocal, the fractional error in distance equals the fractional error in the angle. A 1% angle measurement gives a 1% distance.
  
  $\frac{Δ d}{d} = \frac{Δ p}{p} : p = 0. 1^{''} \pm 0.00 1^{''} \Rightarrow \pm 1%$
5. But the angle shrinks as distance grows while the measurement error does not. Push out to p = 0.001″ with the same ±0.001″ instrument and the fractional error reaches 100% — the measurement no longer says anything.
  
  $p = 0.00 1^{''} \pm 0.00 1^{''} \Rightarrow \pm 100%$
Answer

10 parsecs, 32.6156 light-years. The reciprocal is what makes parallax both exact and short-ranged: error grows in proportion to distance, so every parallax survey has a horizon set by its angular precision rather than by how faint it can see. Hipparcos measured to about 0.001″ and reached a few hundred parsecs; Gaia reaches ~0.00002″ and so gets tens of thousands. Beyond that horizon astronomy must change method entirely — which is why the distance ladder has rungs at all.

$d = \frac{1}{p} \frac{Δ d}{d} = \frac{Δ p}{p}$

Learning path

How far away is it?

Leads to Solar luminosity the distance itself, in parsecs, from triangulation and nothing else — no assumption about the star required.

References (3)

The first stellar parallax ever measured, and the source of insight 1’s 0.31″: Bessel, "Bestimmung der Entfernung des 61sten Sterns des Schwans." Astronomische Nachrichten 16, 65–96, 1839.
The mission the challenge block credits with sub-milliarcsecond astrometry: T. Prusti, J. H. J. de Bruijne, A. G. A. Brown, A. Vallenari, C. Babusiaux & C. A. L. Bailer-Jones, "The Gaia mission." Astronomy & Astrophysics 595, A1, 2016.
The parsec as a defined unit, which is why d = 1/p has no constant in it: International Astronomical Union, Resolution B2 on recommended zero points for the absolute and apparent bolometric magnitude scales. XXIX General Assembly, Honolulu, 2015.

By the 42Math team · Updated 2026-05-21

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Nearby star (large parallax) - p = 0.77″ → 1.2987 pc, or 4.2358 ly: the nearest star system, and even it shifts by less than one arcsecond
Sun-like star at 10 pc - p = 0.1″ → exactly 10 pc and 2062650 AU — a tenth of the angle is ten times the distance, with no constant in between
Distant, faint target - p = 0.02″ → 50 pc, and the readout warns that below about 0.01″ the atmosphere blurs the shift away