Eclipse Fraction Calculator: Find the Sun’s Coverage

This eclipse fraction calculator works out what fraction of the Sun’s diameter is covered by the Moon during a partial solar eclipse. You enter the distance between the centers of the Sun and Moon as seen from your location, and the calculator returns the fraction covered (from 0 to 1) and the corresponding percentage. It uses the standard geometric formula for overlapping circles, so you get an instant, accurate answer without drawing anything.

Formula

The calculator uses the linear overlap fraction along the line connecting the centers:

f = (1 + r – d) / (2r)

• f = fraction of the Sun’s diameter that is covered (0 to 1)
• d = distance between the centers of the Sun and Moon, in units of the Sun’s radius (enter 0 for a total eclipse)
• r = radius of the Moon relative to the Sun (typical value 0.5, meaning the Moon’s apparent radius is half the Sun’s)

The formula assumes the Moon and Sun are perfect circles, which is a very good approximation for eclipse geometry. If d is greater than 1 + r, there is no overlap and the fraction is 0. If d is less than |1 – r|, the smaller body is completely inside the larger, so the fraction is 1 (or r/1 if the Moon is larger, but that does not happen in practice).

Worked Example

Using the defaults: d = 0.5 solar radii, r = 0.5 (Moon radius = half the Sun’s).

1. Compute the numerator: 1 + 0.5 – 0.5 = 1.0
2. Compute the denominator: 2 × 0.5 = 1.0
3. Divide: 1.0 / 1.0 = 1.0
4. Clamp to [0,1]: 1.0 (already in range)
5. Percent = 1.0 × 100 = 100%

So with the default inputs, the calculator shows 100% coverage, which makes sense: when the Moon’s center is only 0.5 solar radii from the Sun’s center and the Moon’s radius is 0.5, the Moon completely covers the Sun (a total eclipse).

When to Use This Calculator

Use this tool when you have a specific geometry for a partial solar eclipse—for example, from a simulation, a table of Besselian elements, or a planetarium app that gives the separation of centers at a given time. It is also handy for:

• Checking the maximum eclipse magnitude for your location during an upcoming partial eclipse.
• Understanding how a small change in the Moon’s path affects the coverage.
• Teaching the geometry of eclipses in a classroom.

Note that this calculator gives the linear fraction of the Sun’s diameter, not the area fraction. The area covered is always less than the linear fraction for partial phases. For example, a 50% linear coverage corresponds to about 39% of the Sun’s area covered.

How to Get the Input Values

You need the apparent distance between the centers of the Sun and Moon, and the apparent radius of the Moon, both in units of the Sun’s apparent radius. Here is how to get them:

1. From an ephemeris or planetarium software: Look for the “center-to-center distance” or “separation” between the Sun and Moon at the time of interest. It is often given in arcminutes or degrees. Divide that by the Sun’s apparent radius (about 16 arcminutes) to get d.
2. From Besselian elements: The values of x and y (in Earth radii) can be converted to apparent separation, but that is complex. Simpler: use the “magnitude” from eclipse predictions—that is exactly the fraction this calculator gives, so you can reverse-check.
3. From a photo: If you have a calibrated image, measure the diameters and the gap between the limbs, then compute d as (Sun radius + Moon radius – overlap) / Sun radius.

For most people, the easiest path is to use an eclipse prediction site that lists the “magnitude” at maximum eclipse for their location—that number is exactly the fraction this calculator outputs. You can then use the calculator to explore how the fraction changes at other times.

Common Mistakes and Misconceptions

A frequent error is confusing the fraction of the diameter with the fraction of the area. The media often report “85% of the Sun covered” meaning diameter, but the actual area blocked is smaller. For a linear fraction f, the area fraction is approximately f² (for small f) but for large f it is closer to 1 – (1-f)². For f=0.85, the area fraction is about 0.81, so the difference is noticeable.

Another mistake is using the Moon’s radius relative to the Sun as 1.0. In reality, the Moon’s apparent radius is always smaller than the Sun’s (typically 0.5 to 0.6). If you set r = 1, you are modeling two identical circles, which is not an eclipse.

Also, the calculator assumes the Moon is completely inside the Sun when d < |1 - r|. For a total eclipse, d = 0 and r < 1, so the fraction is 1. That is correct.

Limits of the Formula

The formula is purely geometric and ignores:

• Atmospheric refraction, which can slightly shift the apparent positions of the Sun and Moon near the horizon.
• Non-circular shapes—the Sun and Moon are not perfect circles, but the difference is negligible for eclipse purposes.
• Lunar limb irregularities—mountains on the Moon can cause Baily’s beads, but the overall fraction is unaffected.

For precise predictions, use official eclipse data. This calculator is for educational and planning purposes, giving you a quick estimate of coverage.

Frequently Asked Questions

What is the difference between magnitude and obscuration?

Magnitude is the fraction of the Sun’s diameter covered by the Moon (what this calculator gives). Obscuration is the fraction of the Sun’s area covered. Obscuration is always less than magnitude for partial eclipses. For example, a magnitude of 0.8 gives an obscuration of about 0.74.

How do I find the distance between centers for my location?

Use a planetarium app like Stellarium or an eclipse prediction website. At the time of maximum eclipse, they often list the “magnitude”—that is your fraction. To get the distance, you can work backward: d = 1 + r – 2r × magnitude. Or, if you have the separation in arcminutes, divide by the Sun’s radius (about 16 arcminutes).

Can this calculator predict the exact time of an eclipse?

No. It only computes the fraction for a given geometry. To find the time, you need an ephemeris that gives the positions of the Sun and Moon over time. You can then use this calculator at each time to see when the fraction peaks.

Why does the calculator show 100% for a total eclipse even if the Moon is smaller?

Because the fraction is defined as the portion of the Sun’s diameter that is covered. When the Moon is completely inside the Sun’s disk, the entire diameter is covered, so the fraction is 1.0 (100%). The Moon being smaller only affects the duration of totality, not the maximum fraction.

Is it safe to look at the Sun during a partial eclipse?

No. Never look directly at the Sun without proper eye protection, such as eclipse glasses or a solar filter. Even when 99% covered, the remaining sliver is intense enough to cause permanent eye damage. Use certified solar viewers.

What if the Moon is larger than the Sun in the sky?

That happens during a total eclipse when the Moon is near perigee. In that case, the Moon completely covers the Sun, and the fraction is still 1.0. The formula handles it because r > 1, but the output is capped at 1.0.

Related

• Fraction Calculation Questions: How to Solve Them
• Ounces to Pounds: Formula & Easy Converter
• 3 Liters to Ounces Calculator: Quick Conversion

in d | Center-to-center distance | solar radii | 0.5 | 0
in r | Moon radius relative to Sun | 0.5 | 0.1
out fraction | Fraction of Sun's diameter covered | - | max(0, min(1, (1 + r - d) / (2 * r))) | 3
out percent | Percent of Sun's diameter covered | % | max(0, min(1, (1 + r - d) / (2 * r))) * 100 | 1