Is there a formula to calculate the area that the bead on a rifle sight covers at a given range?
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Is there a formula to calculate the area that the bead on a rifle sight covers at a given range?
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It will be different depending on who is holding the gun. You have a fixed size object that will be at different length depending on how you hold the gun. Different people will hold the gun differently and maybe a little further forward or back.
For example, if you hold you hand at arms length and use your thumb to blot out, say a car. If you move your hand closer to you, you may blot out 23 cars. There is no way to predict how much it will cover without very precise measurements as to how you hold the gun and if you hold it exactly the same every time.
That being said, many red dot sights are 4 MOA (minute of angle). (A minute of angle is 1" at 100yds. Some people try to dispute this and say that is not exact, demonstrating a very discouraging lack of geometric education. If you have a equilateral triangle of 100 yds on all sides, there are 3600 inches on each side of that triangle. An equilateral triangle has each corner set at 60 degrees.3600 divided by the 60 degrees brings it down to 60 inches in one degree at 100 yards. There are 60 minutes in one degree. 60 inches divided by 60 minutes are exactly one minute of angle at 100 yds)Sorry for the digression, but that is a pet peeve of mine.
A red dot sight with a 4 MOA dot will cover a 4"circle at 100 yds, 8" at 200, etc. It doesn't sound like a lot until you figure, with a decent gun and scope you have a lot better for marksmanship score than 4" margin of error. I have a rifle that I have put 3 rounds, covered with a nickel, no hole showing at 100 yds.
If you have a large bead sight at the muzzle it will make it harder to get good, tight groups, but the smaller it is the easier it can be damaged. If it gets knocked off or bent, it is worthless. and so is your gun until you can get a scope on it.

To be quiet honest with you , i never was very good with math,but shooting at targets the front bead is set between the rear sight wedge and this sight picture is held on the target ,sometimes too much thinking and not enough time at the range leaves one too wonder.
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If I remember my trig/radial geometry correctly, one minute of angle at 100yds is about 1.05"  you're looking at the tangent as it appears flat on the target surface, outside edge of the circle, at 100 yards. (Tangent of 1 degree = 0.0175, b = 3,600 inches, one minute is 1/60th of a degree.).
And, yes, there is a way to estimate the area blotted out. It will only be close. Measure the width of the front bead. Use a caliper if you have one. Use inches to as close a decimal value as you can. If you use fractions convert to decimals. This is value a.
Then have someone measure from your eyeball to the bead (inches, again). This is value b. For the math majors we are assuming a very small right angle for practicality.
Divide the bead width (a) by the distance from eye to bead (b). That gives you the value for what is called Tan A (tangent A in Trig).
For whatever distance you want to figure out how much is blocked, multiply that distance, in inches, by Tan A.
For example, if the bead is 1/8 inch wide (0.125 inches), and the distance to your eye is 30 inches, then the value of Tan A is 0.00417 (0.125 / 30 = 0.00417). For a distance of 50 yards (50 yards x 36 inches/yard = 1,800 inches), the bead would block out about 7.5 inches (1,800 x 0.00417).
Someone want to double check my trig. It's been awhile.
More practical is doing this backward from a target to figure out how far to drift a front site and to raise or lower it.
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