Angular Resolution Calculator for telescopes, lenses, eyes and radio dishes. Solve the Rayleigh limit, aperture or wavelength, then find the smallest detail you can split at range.
Green light (550 nm) suits visual use. Radio users can type the frequency in MHz or GHz.
Target & Focal Length
Step-by-Step Working
- λ = 550 nm = 5.5 × 10⁻⁷ m
- D = 10 cm = 0.1 m
- sin θ = 1.22 × λ ÷ D = 1.22 × 5.5 × 10⁻⁷ ÷ 0.1 = 6.71 × 10⁻⁶
- θ = arcsin(6.71 × 10⁻⁶) = 6.71 × 10⁻⁶ rad
- θ = 6.71 × 10⁻⁶ × 206,264.806 = 1.384 arcsec
Resolution by Aperture
| Aperture | Rayleigh (arcsec) | Dawes (arcsec) | Detail at Target (in) |
|---|---|---|---|
| 50 mm (1.969 in) | 2.768 | 2.32 | 0.8503 |
| 70 mm (2.756 in) | 1.977 | 1.657 | 0.6074 |
| 80 mm (3.15 in) | 1.73 | 1.45 | 0.5314 |
| Yours · 100 mm (3.937 in) | 1.384 | 1.16 | 0.4251 |
| 102 mm (4.016 in) | 1.357 | 1.137 | 0.4168 |
| 127 mm (5 in) | 1.09 | 0.9134 | 0.3348 |
| 152 mm (5.984 in) | 0.9106 | 0.7632 | 0.2797 |
| 203 mm (7.992 in) | 0.6818 | 0.5714 | 0.2094 |
| 254 mm (10 in) | 0.5449 | 0.4567 | 0.1674 |
| 305 mm (12.01 in) | 0.4538 | 0.3803 | 0.1394 |
| 406 mm (15.98 in) | 0.3409 | 0.2857 | 0.1047 |
| 2,400 mm (94.49 in) | 0.05767 | 0.04833 | 0.01771 |
Resolution by Wavelength
| Wavelength | Band | Rayleigh (arcsec) | Detail at Target (in) |
|---|---|---|---|
| 365 nm | Ultraviolet | 0.9185 | 0.2821 |
| 450 nm | Visible (blue) | 1.132 | 0.3478 |
| Yours · 550 nm | Visible (green) | 1.384 | 0.4251 |
| 656.3 nm | Visible (red), H-alpha | 1.652 | 0.5073 |
| 1.25 µm | Infrared (J band) | 3.146 | 0.9662 |
| 2.2 µm | Infrared (K band) | 5.536 | 1.701 |
| 10 µm | Infrared (mid-IR) | 25.16 | 7.73 |
| 1 mm | Microwave (mm) | 2,516 | 773 |
| 21.11 cm | Microwave (21 cm H I) | None | None |
Angular Resolution Calculator Formula
The angular resolution calculator uses the Rayleigh criterion. Two point sources count as resolved when the bright center of one diffraction pattern sits on the first dark ring of the other. For a round lens or mirror, that spacing sets the smallest angle the optics can split.
The angular resolution calculator solves the exact form of the rule, with the sine of the angle on the left side. It reads the wavelength and aperture you enter and returns the smallest angle they can split.
$$\sin\theta = 1.22 \frac{\lambda}{D}$$
Here θ is the angular resolution, λ is the wavelength and D is the aperture diameter. λ and D must share one length unit before you divide. The factor 1.22 comes from the first zero of the Bessel function J1, which is 3.8317, divided by π.
For a telescope, camera lens or eye, sin θ and θ are almost the same number. That is why most textbooks and the HyperPhysics Rayleigh criterion page use the short form below.
$$\theta \approx 1.22 \frac{\lambda}{D}$$
The exact form only matters when the aperture is a few wavelengths wide, such as a small radio antenna. If 1.22λ/D comes out above 1, no angle exists. The angular resolution calculator then asks for a larger aperture or a shorter wavelength.
Worked Example: a 10 cm Telescope
A common question asks for the angular resolution of a 10 cm diameter telescope. Use green light at 550 nm, which is the tool’s default. Convert both values to meters first: λ = 5.5 × 10⁻⁷ m and D = 0.1 m.
$$\theta = 1.22 \times \frac{5.5 \times 10^{-7}}{0.1} = 6.71 \times 10^{-6}\ \text{rad}$$
Multiply radians by 206,264.806 to get arcseconds. The result is 1.384 arcseconds, or 6.71 microradians. This is the figure the hero card shows when the tool first loads.
At one mile, that angle spans about 0.425 inches. On the Moon it spans about 2.58 km (1.6 mi). Any two craters closer together than that blend into one blob in a 10 cm telescope.
Solving for Aperture or Wavelength
Switch Solve For to Aperture Diameter to find the lens or mirror size a target angle needs. A 1 arcsecond goal at 550 nm needs about 13.84 cm (5.45 in) of clear aperture.
$$D = \frac{1.22 \lambda}{\sin\theta}$$
The Wavelength mode returns the longest wavelength a given aperture can use and still split your target angle. Any longer wave spreads the Airy disk too wide for that detail.
$$\lambda = \frac{D \sin\theta}{1.22}$$
What Angular Resolution Measures
Angular resolution, also called resolving power, is the smallest angle between two points that an instrument can still show as two. It is an angle, not a length. The same 1.4 arcsecond limit applies to a double star, a lunar crater or a road sign a mile away.
A smaller angle is better, because it means finer detail. A telescope rated at 0.5 arcseconds shows more than one rated at 2 arcseconds. People call the smaller angle “higher resolution,” which is why both “higher” and “lower” turn up in questions about it.
Angular distance, also called angular separation, is the angle between two objects as seen by the viewer. The angular resolution is simply the smallest angular distance the optics can still split.
Units of Angular Resolution
Astronomers quote arcseconds, optical engineers often use microradians, and radio astronomers use arcminutes or degrees. The Result Unit menu in the tool switches between all six. The table gives the conversion factors for converting angular sizes by hand.
| Unit | Degrees | Arcminutes | Arcseconds | Microradians |
|---|---|---|---|---|
| 1 radian | 57.2958 | 3,437.75 | 206,264.8 | 1,000,000 |
| 1 degree | 1 | 60 | 3,600 | 17,453.3 |
| 1 milliradian | 0.0573 | 3.438 | 206.26 | 1,000 |
| 1 arcminute | 0.01667 | 1 | 60 | 290.89 |
| 1 arcsecond | 0.000278 | 0.01667 | 1 | 4.848 |
One arcminute is the same as one MOA, so shooters can carry these figures straight into the MOA calculator. For general angle work in degrees, radians and revolutions, the angular displacement calculator handles the conversion.
Rayleigh vs Dawes Limit
Rayleigh’s rule comes from the physics of diffraction. Dawes’ limit is an observing rule. William Rutter Dawes found it by testing many telescopes on double stars, as the Orwell Astronomical Society describes.
$$R = \frac{116}{D}$$
R is the resolving power in arcseconds and D is the aperture in millimeters. With D in inches, the constant becomes 4.56. Sky & Telescope notes that it holds for two equally bright stars of about 6th magnitude.
At 550 nm the Rayleigh form works out to about 138/D, so Dawes is roughly 16% tighter. A trained eye can see a slight dip between two touching star images, which Rayleigh’s rule does not count. The angular resolution calculator shows Dawes only for visible light, since it was found by eye.
Telescope Resolution by Aperture
This table works as a quick telescope resolution calculator for common sizes at 550 nm. The Moon column uses NASA’s mean Earth-Moon distance of 384,400 km from the NASA Moon Fact Sheet.
| Aperture | Inches | Rayleigh (arcsec) | Dawes (arcsec) | Smallest Detail on the Moon |
|---|---|---|---|---|
| 50 mm | 1.97 | 2.77 | 2.32 | 5.16 km |
| 70 mm | 2.76 | 1.98 | 1.66 | 3.68 km |
| 80 mm | 3.15 | 1.73 | 1.45 | 3.22 km |
| 102 mm | 4.02 | 1.36 | 1.14 | 2.53 km |
| 127 mm | 5.00 | 1.09 | 0.913 | 2.03 km |
| 152 mm | 5.98 | 0.911 | 0.763 | 1.70 km |
| 203 mm | 7.99 | 0.682 | 0.571 | 1.27 km |
| 254 mm | 10.0 | 0.545 | 0.457 | 1.02 km |
| 2.4 m (Hubble) | 94.5 | 0.0577 | 0.0483 | 0.107 km |
Doubling the aperture halves the angle in every row. The Resolution by Aperture panel in the tool builds the same table for your own wavelength and target distance, with your aperture highlighted.
Reading the Result Cards
Past the headline angle, the angular resolution calculator turns the result into four practical answers. Each card ties the angle to something you can test at the eyepiece, on a sensor or in a spec sheet.
Detail at Target
This card answers how to calculate spatial resolution, meaning the smallest real-world size you can split at a set distance. Open Target and Focal Length to enter that distance in feet, miles, kilometers, AU or light-years.
$$s = 2L \tan\left(\frac{\theta}{2}\right) \approx L\theta$$
L is the distance to the target and θ is in radians. The card also gives the size on the Moon and the range at which a 1 inch (or 1 cm) detail stays resolved. For the 10 cm default, a 1 inch detail stays split out to about 2.35 miles.
Airy Disk Diameter
The central bright spot, the Airy disk, spans twice the Rayleigh angle from one dark ring to the other. For the 10 cm default that is 2.77 arcseconds across. The Dawes row sits beside it for visual double-star work.
The last row sizes the disk where it lands on a camera sensor or film. N is the focal ratio, the focal length divided by the aperture.
$$d_{Airy} = 2.44 \lambda N$$
A 900 mm focal length on a 100 mm aperture gives f/9 and a 12.08 µm disk at 550 nm. If your sensor pixels are as large as that disk, the pixels set the limit rather than the optics.
Magnification to See It
A sharp eye splits about 1 arcminute, or 60 arcseconds. The telescope must enlarge the resolved detail up to that size before you can see it. Dividing 60 by the Rayleigh angle in arcseconds gives the lowest magnification that shows the full limit.
$$M = \frac{60}{\theta}$$
For the 10 cm default that is about 43×. A telescope’s angular magnification equals the objective focal length divided by the eyepiece focal length. So a 900 mm telescope with a 20 mm eyepiece gives 45×, just enough.
$$M = \frac{f_{obj}}{f_{eye}}$$
The same card compares your result with the 2.4 m Hubble mirror at the same wavelength. It also lists the aperture needed to reach 1 arcsecond. When you enter 2.4 m yourself, the comparison switches to the 6.5 m JWST mirror.
Frequency and Photon Energy
The fourth card converts between wavelength and frequency, names the spectrum band and gives the photon energy. Green 550 nm light works out to 545.1 THz and 2.254 eV.
$$f = \frac{c}{\lambda} \qquad E = \frac{hc}{\lambda}$$
Resolution grows coarser in step with wavelength, so the band matters as much as the aperture. For more work on these links, the wavelength calculator and the frequency calculator cover the wave side in detail.
Human Eye Angular Resolution
The pupil is the eye’s aperture, so the same formula gives the eye’s diffraction limit. The pupil shrinks in bright light and opens to about 7 mm when dark-adapted, according to Roger N. Clark’s notes on the human eye.
| Pupil Diameter | Arcseconds | Arcminutes | Microradians |
|---|---|---|---|
| 2 mm | 69.2 | 1.15 | 336 |
| 3 mm | 46.1 | 0.77 | 224 |
| 5 mm | 27.7 | 0.46 | 134 |
| 7 mm | 19.8 | 0.33 | 95.9 |
Real vision is coarser than these figures, because the retina and the eye’s lens add their own limits. Normal 20/20 vision resolves detail of about 1 arcminute. HyperPhysics notes that the sharpest human vision comes within about a factor of two of the diffraction limit.
Enter your pupil size as the aperture to compare with any telescope. Against a 2.4 m mirror at 550 nm, a 3 mm pupil is coarser by a factor of 800.
Is the Human Eye 576 Megapixels?
The 576 megapixel figure is Roger Clark’s estimate. He takes a pixel spacing of 0.3 arcminute, close to the eye’s finest acuity, and spreads it over a 120 by 120 degree view. That gives 24,000 by 24,000 pixels.
Clark himself says the eye does not see that many pixels at once. The eye moves and builds the detail over time, more like a video stream than one snapshot. The figure is the detail an image would need to match what you can see by scanning a scene.
Can the Human Eye See in 4K?
Yes, at a close enough distance. A 65 inch 16:9 4K screen has pixels about 0.0148 inches wide. At 1 arcminute acuity, those pixels blend together beyond about 4.2 feet.
A 1080p screen of the same size has pixels twice as wide, which blend beyond about 8.5 feet. Between those two distances, a 20/20 viewer can see the extra 4K detail. Past 8.5 feet the two screens look the same in sharpness.
How to Improve Angular Resolution
The formula has only two levers. A larger aperture or a shorter wavelength gives a smaller angle. Doubling the aperture halves the angle, so bigger telescopes do resolve finer detail, all else being equal.
From the ground, the air sets a second limit called seeing. The Isaac Newton Group of Telescopes reports that seeing is rarely below 0.5 arcseconds even at the best sites. That is about ten times coarser than the diffraction limit of a 4 m telescope.
So a good resolution for a backyard telescope is one that matches the sky. At 550 nm, about 14 cm of aperture reaches 1 arcsecond. The angular resolution calculator flags results finer than 1 arcsecond with a seeing warning for that reason.
Space telescopes and adaptive optics remove the seeing limit. Interferometers go further by linking separate telescopes, where the baseline B between them takes the place of the aperture.
$$\theta \approx \frac{\lambda}{B}$$
A 1.3 mm radio wave across a 10,000 km baseline works out to about 27 microarcseconds. That is why interferometer arrays hold the finest angular resolution of any telescope type.
Radio Telescope Resolution
Radio users can type the frequency in MHz or GHz in the wavelength field. The tool converts it with λ = c/f and switches its tables to dish sizes from 1 m to 305 m.
The 21 cm hydrogen line at 1,420.4 MHz shows why radio dishes must be huge. A 25 m dish resolves only about 35.4 arcminutes, wider than the full Moon. A 100 m dish reaches about 8.85 arcminutes.
At radio wavelengths the dish may be only a few hundred wavelengths wide. The exact sine form in the angular resolution calculator keeps those results correct. For transmit power and dish gain, the EIRP calculator covers the link-budget side.
Angular Resolution Calculator FAQs
These answers cover the questions people search most often about angular resolution. Each one uses the same Rayleigh formula the tool runs, with numbers you can check in the tool.
How do you calculate resolving power?
Divide 1.22 times the wavelength by the aperture diameter, in the same units, to get the angle in radians. Multiply by 206,264.8 for arcseconds. For visual double stars, divide 116 by the aperture in millimeters to get the Dawes limit in arcseconds.
A 203 mm telescope gives 0.68 arcseconds by Rayleigh at 550 nm and 0.57 arcseconds by Dawes. Both describe the same telescope; Dawes simply counts a fainter dip between two star images as a split.
Can a 70mm telescope see Jupiter?
Yes. A 70 mm telescope resolves about 1.98 arcseconds at 550 nm. Jupiter spans 30.5 to 50.1 arcseconds from Earth, according to the NASA Jupiter Fact Sheet.
That gives roughly 15 to 25 resolution elements across the disk. It is enough to show the two main cloud belts and the four large moons as points. Finer detail inside the belts needs a larger aperture and steady air.
What are the four types of resolution?
In remote sensing and imaging, the four types are spatial, spectral, temporal and radiometric resolution. Spatial resolution is the smallest feature size, which is the Detail at Target card in this tool.
Spectral resolution is how finely the wavelength is split, and temporal resolution is how often an image is taken. Radiometric resolution is how many brightness levels the sensor records. Angular resolution feeds directly into spatial resolution once you know the distance.
Is a smaller diffraction limit better?
Yes. A smaller diffraction limit means a smaller angle, which means finer detail. The diffraction limit is the best case, though, set only by the aperture and the wavelength.
Lens flaws, poor focus, air turbulence and large sensor pixels can only make the real result coarser. A telescope that reaches its diffraction limit in practice is said to be diffraction limited.
What is the minimum angular resolution?
There is no single minimum, because it depends on the instrument and the wavelength. A sharp eye reaches about 1 arcminute, a 10 cm telescope about 1.38 arcseconds, and Hubble about 0.058 arcseconds in green light.
Radio interferometers with baselines thousands of kilometers long reach tens of microarcseconds. At the other end, an aperture only 1.22 wavelengths wide spreads light over the full 90 degrees and resolves nothing.
What is the formula for limit of resolution?
The angular limit of resolution is θ = 1.22 λ/D, from the Rayleigh criterion. At the focal plane, the matching linear limit is the Airy disk radius, which equals 1.22 λ N for a focal ratio N.
For example, at f/9 and 550 nm the radius is about 6.04 µm. Two star images closer than that on the sensor merge. Browse the full set of optics calculators for lens and wave tools that build on this limit.