Why does a rainbow appear semi-circular?

Why is a rainbow semi-circular?

A rainbow is an optical (and meteorological) phenomenon that is caused by both reflection and refraction of light in water droplets in the Earth's atmosphere, resulting in a dispersed light of different constituent colours appearing in the sky. It takes the form of a multi-coloured arc. Rainbows caused by sunlight always appear in the section of sky directly opposite the sun.

The colours of the rainbow are violet, indigo, blue, green, yellow, orange and red. An easy way to remember the colours of the rainbow is VIBGYOR.

It is believed that rainbow got its name from its shape. A rainbow's arc looks similar to a bow. People also noticed that this colourful arc would only form when it was raining so they called it a rainbow.

Scientist Isaac Newton explained how a rainbow is formed. Sunlight is made up of the colours of the rainbow. When those colours are all mixed together it appears white. White light is the light we see every day. When sunlight travels through the air in the same direction we see white light. When the sunlight enters through a raindrop the light is first refracted into raindrop, thereafter, within the raindrop total internal reflection (which is a form of refraction) occurs. It is followed by reflection within the raindrop and then the light is refracted out of the raindrop through the opposite surface. At each stage of refraction the colours get separated out because of their different wave lengths and different energy levels. These coloured rays of optical energy refract through different angles and are separated out. Each raindrop makes its own rainbow but it takes millions of raindrops for us to see a rainbow. The finally reflected colours fall on the dust and gaseous particles in the atmosphere. Since these are at a far distance, our eyes cannot fathom (make out) the different height at which the different colours of light strike. We get an illusion as if the rainbow is in a semi-circular shape.  

A rainbow arc is a 420  angle starting from the direction opposite from the sun. (This is because the general angle of refraction is about 420.

Sometimes sunlight is reflected twice inside a raindrop. When this occurs we get a secondary rainbow or a double rainbow. The colours of a secondary or double rainbow are in opposite order of the primary or first rainbow. You will see violet on top of the secondary rainbow instead of red.

Moon bows or lunar rainbows occur when the moon's light reflects through the raindrop. This type of rainbow is rare because the moon's light usually isn't bright enough for a rainbow to form.

All rainbows are full circles, however, the average observer only sees approximately the upper half of the arc: 'the illuminated droplets above the horizon from the observers line of sight'.

The bow is centred on the shadow of the observer's head, and forms a circle at an angle of 40–42° to the line between the observer's head and its shadow. As a result, if the Sun is higher than 42°, then the rainbow is below the horizon and usually cannot be seen as there are not usually sufficient raindrops between the horizon (that is: eye height) and the ground, to contribute. Exceptions occur when the observer is high above the ground, for example in an aeroplane. Alternatively, you might see the full circle in a fountain or waterfall spray if you are at the right point. Mountain climbers may sometimes see more of a full-circle rainbow. Even a circular rainbow is an illusion. The illusion is caused because the colours of the rainbow bounce (reflect) from great distances and we feel as if they are bouncing from a circular surface.