Perform fast calculations with our user-friendly online calculator! Conveniently crunch numbers and solve equations instantly. Ideal for quick math tasks, our tool simplifies your daily computations effortlessly. Try our intuitive calculator for accurate results on the go!
Calculate angular resolution fast. Use our clear formula and steps to find resolution in radians, arcseconds, and degrees. Built for students and hobbyists.
Angular resolution tells you how close two points can be and still look like two. It shows the smallest angle an optical system can separate. You’ll use it for telescopes, microscopes, and antenna arrays. This guide keeps math simple. You’ll see the formula, steps, and a worked example. Use our calculator to get quick results.
Angular resolution is an angle. It says how sharp an image appears. A small angle means you see more detail. A big angle means less detail. For a single circular aperture, the key is diffraction. The Airy pattern sets the limit. For arrays, the baseline sets the limit.
Rayleigh criterion for a circular aperture: θ = 1.22 × λ / D
Interferometer (baseline) approximation: θ ≈ λ / B
In those lines, θ is in radians. λ is wavelength in meters. D is aperture diameter in meters. B is baseline in meters. To get arcseconds, multiply radians by 206,265. To get degrees, multiply radians by 180 / π.
You enter wavelength and aperture or baseline. The script converts units to meters. It applies the Rayleigh or baseline formula. It returns θ in radians.
Then it converts θ to arcseconds, arcminutes, degrees, milliradians, and microradians. The tool also shows step-by-step math. You see every conversion and divide. The code formats small numbers nicely.
It uses scientific style for very tiny values.
Step 1. Convert wavelength to meters.
Step 2. Convert aperture or baseline to meters.
Step 3. If using aperture, compute numerator = 1.22 × λ.
Step 4. Divide numerator by aperture D to get θ in radians.
Step 5. If using baseline, divide λ by B to get θ.
Step 6. Convert radians to arcseconds: arcsec = θ × 206,265.
Step 7. Convert arcsec to arcmin by dividing by 60.
Step 8. Convert radians to degrees: degrees = θ × (180 / π).
Step 9. Format results for display.
Use visible light at 550 nm and a 0.20 m telescope.
Convert 550 nm to meters. 550 nm = 550 × 10⁻⁹ m = 5.5 × 10⁻⁷ m.
Apply Rayleigh: θ = 1.22 × λ / D.
Numerator = 1.22 × 5.5 × 10⁻⁷ = 6.71 × 10⁻⁷.
Divide by D: θ = 6.71 × 10⁻⁷ / 0.20 = 3.355 × 10⁻⁶ radians.
To arcseconds: 3.355 × 10⁻⁶ × 206,265 ≈ 0.692 arcsec.
So the 0.2 m scope gives about 0.69 arcsec resolution at 550 nm.
Read the results. You’ll see radians, degrees, arcseconds, and more. You’ll also see the math steps. Use those to learn or to verify the result.
Yes. Bigger apertures cut the angle. The Rayleigh formula has D in the denominator. A larger D gives a smaller θ. A smaller θ means finer detail. In short, bigger scopes can see finer detail. That is true if atmosphere and optics are good.
A typical human eye can resolve about 1 arcminute. That is 60 arcseconds. In radians, that is about 2.9 × 10⁻⁴ radians. Many factors change this number. Light level, contrast, and age all matter. But 1 arcminute is a useful rule of thumb.
Lower numeric θ is better. A small angle means you separate close features. So lower is finer. People sometimes say "higher resolution" to mean better detail. That phrase refers to smaller θ or to more pixels. To be precise, smaller angular resolution value equals better optical resolving power.
Angular resolution is simple to compute. The Rayleigh and baseline formulas do the heavy lifting. Use meters for units. Convert results to arcseconds if you work in astronomy. Our calculator does the math fast. It also shows steps so you can learn. Try different wavelengths and apertures. You’ll see how size and light change what you can resolve.
Use θ = 1.22 × λ / D for a circular aperture. For arrays, use θ ≈ λ / B. Convert units to meters first. Multiply radians by 206,265 to get arcseconds.
Use meters. If you use nm or mm, convert them before applying the formula. The calculator handles common unit multipliers.
No. The 1.22 factor comes from the Airy pattern for a circular aperture. It stays the same for ideal, diffraction-limited optics.
Yes. The atmosphere often blurs images. This is why ground telescopes rarely reach their ideal resolution without adaptive optics.
Yes. Use the radio wavelength in meters. For arrays, use baseline B in meters. The same formulas apply.
Astronomers use arcseconds. Small angles map well to arcseconds for stars and planets. It’s the common unit in astronomy.