I need to calculate the visual angle spanned by the moon?

Answer 1

Crikey, that's awesome, Raymond.

However, I would simply stick my outstretched hand in the sky and work out how many moons could stretch across my palm.

My outstretched palm gives me 10 deg of sky.

Also, I already knew the moon was around half a degree across - same as the sun.

Answer 2

Take the diameter of the Moon, divide it by the distance of the observer to the Moon's centre.

You may have to consider the position of the Moon on its orbit (perigee? -- the point closest to Earth -- or apogee? -- the point furthest away)

You may also have to consider the position of the observer. If the observer is right underneath the Moon (i.e., the Moon is at the observer's zenith) you may want to subtract Earth's radius.

Depending on how accurate you want the reading.

The result of the division (Moon Diameter / distance) gives you the angle in radians. To find it in degrees, you can multiply by 180 and divide by pi (there are pi radians in 180 degrees).

---

Moon diameter = 3475 km
Mean distance (centre-to-centre) 384,400 km
Equatorial radius of Earth = 6378
(If the Moon is near the zenith, then the observer must be closer to the equator than to the pole)

Next apogee: Apr. 3 (406,329 km)
Next perigee: Apr. 17 (357,156 km)
(both are centre-to-centre)

The next perigee is very close to the Full Moon.
To calculate the apparent diameter of the full moon as the moon rises, we can use centre-to-centre distance.

3475 / 406,329 = 0.0085522 radian = 0.49 deg. = 29.4'

This will be the smallest Full Moon in 2007.

---

To do it with 'conventional' trigonometry:

Draw a line from the observer to the centre of the Moon. In our example above, this line measures 406,329 km.

At the Moon's centre, the Moon's radius makes an angle of 90 degrees, giving us a right-angle triangle where the small side is 1737.5 km (half the diameter). The third side of the triangle is from the rim of the moon, back to the observer (the hypothenuse).

The angle subtended by the small side (the Moon's radius) at the observer is A.
We know that Tan(A) = opposite side / adjacent side = 1737.5 / 406,329 = 0.0042761...
Therefore A = arcTan (0.0042761...) = 0.245 degree = 14.7'

But you want the apparent angle of the whole diameter (twice the radius) = 2*A = 2*14.7' = 29.4'

-----

In Astronomy we can take one of two shortcuts:

1) use radians (which normally are used for lengths along the circumference) because the circumference of a circle with a diameter of 406,329 km looks almost like a straight line over such a 'short' section (3475 km)

2) Place the right angle along one rim instead of at the centre. At that distance, it will not make a big difference.
e.g., Tan(2A) = 3475 / 406,329 = 0.0085522
(note that this is the same number as the angle in radian).
Then 2A = arcTan(0.0085522) = 29.3995'
(a difference of 0.0005' = an 'error' of less than 0.002%)

For very small angles, these approximations are OK:
Sin(A) = A in radian,
Tan(A) = A in radian
This means that it does not matter too much where you put the right angle on the Moon, you'll still get a good approximation.

Answer 3

Raymond's math is mostly right, however one should use the sine instead of the tangent. Tangent seems to make sense if you consider the moon a flat disc, but you also have to remember it really is a sphere with depth. Sine and Tangent generally don't give very different answers in this case, but the reason why the Sine is more appropriate is illustrated in this picture: http://public.clunet.edu/~sjfahmie/angles.gif

To demonstrate the discrepancy, calculate the angle subtended by the Earth from the perspective of someone standing on it. Since we are standing on the surface, the ratio of our distance to the center of the Earth and it's radius is approximately 1. 2*arctan(1) = 90 degrees, where 2*arcsin(1) is 180 degrees. Obviously 180 degrees is correct.