Angular size calculator
If you know how far away an object is and what angle it spans in your field of view, you can work out its real size with basic trigonometry. Enter both and the calculator draws the triangle and gives the result.
Key formula
size = 2 · d · tan(θ / 2) · for small angles: size ≈ d · θ (θ in radians)
Try it
How to use the simulator
If you know how far away an object is and the angle it spans in your field of view, you can work out its real size with basic trigonometry: size = 2 · d · tan(θ/2). It's the same method astronomers use for lunar craters or planetary diameters. The diagram shows the triangle formed by the observer and the two ends of the object.
Controls
- Distance: 10 to 1,000 m.
- Angle: 1° to 90°, to a tenth of a degree.
Things to try
- Check Grace's calculation: at 217 m and 35.44° the object comes out at about 139 m.
- Double the distance without touching the angle: the size doubles too. A distant object and a close one can look equally big.
- Change the angle by just 1°. At 217 m that's several meters, so measuring the angle well matters as much as measuring the distance.
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The observer's triangle
Two lines run from your eye, one to each end of the object. They form an isosceles triangle with its tip at you. Split it in half and you get two right triangles, and in each one the tangent of half the angle equals half the size divided by the distance.
For small angles, under about 10°, the tangent is very close to the angle itself in radians, and the formula simplifies to size ≈ distance · angle. Astronomers use that approximation almost all the time, because objects in the sky span tiny angles.
Measuring angles without instruments
At arm's length, your little finger covers about 1°, your fist about 10° and your spread hand from thumb to little finger about 20°. The full Moon spans half a degree: your little finger covers it easily.
The same method works in reverse: if you know the real size and measure the angle, you get the distance. That is how distances are estimated in surveying, navigation and artillery.
Worked example
Example: the Moon's diameter
- The Moon is about 384,400 km away and spans about 0.518°.
- Size = 2 · 384,400 · tan(0.259°) = 2 · 384,400 · 0.00452 ≈ 3,475 km, its real diameter.
- At human scale: a building 200 m away that spans 10° is 2 · 200 · tan(5°) ≈ 35 m tall.
- Change the angle by just 1° and the result moves by about 3.5 m: measuring the angle well matters as much as measuring the distance.
In Project Hail Mary
In the novel, the main character works out the size of an unknown object in space from its distance and the angle it spans in an image. With 217 m and 35.44° it comes out at about 139 m.
Frequently asked questions
Why do the Sun and Moon look the same size?
By coincidence. The Sun is about 400 times wider than the Moon, but also about 400 times farther away, so both span about half a degree. That is what makes total solar eclipses possible.
What is an arcsecond?
One 3,600th of a degree. Telescopes work in arcseconds: Jupiter, for example, spans 30 to 50 arcseconds as seen from Earth.
Why does the Moon look bigger near the horizon?
It is an optical illusion. Measured, it spans the same angle on the horizon as high in the sky; it is your brain that reads it as larger.