How do lenses work?

There are a few things that you need to know about lenses in order to understand and to use them.

We will only use concave lenses because those are the one that allow you to create an image. First we'll discuss the parts of a lens. Here is a drawing of a normal concave lens:

F is called the focal point of the lens and r is the radius of the circle that makes up a lens.
Now, there are a few special ways that a beam of light can hit the lens: straight in the lens, through the focal point or through the middle of the lens. If a beam of light hits the lens in one of these ways, it will be broken in a special way. Look at the picture below:

Knowing this you can draw an image as it would be projected by using 2 light beams that hit the lens in one of the above ways (in drawings, we use a line with two arrows pointing outwards as a concave lens and, arrow as an object and lines with arrows on them as beams of light).

The grey arrow on the left is the object, the grey arrow on the right is the image.

Now that you know how lenses work, we can begin with the basic lens formula witch is:

Where f stands for focal length, o for the distance between lens and object and i for the distance between image and lens.

Another formula is the magnification formula:

M stands for magnification and you already know what o and i mean. If M is negative, the image is flipped upside down. If M is larger than 1, the image is M times bigger than the object (for example if M=2 the image is twice as large). If M is smaller than 1, the image is M times smaller than the object (for example if M=1/2 or 0.5 the image is half as big as the object). So if you need to know how large your projected image will be even before you have bought anything, you can use this formula. Also it's best to use this formula and the above to decide which focal length you are going to use for your fresnel lens.

With these two formulas you can calculate everything you need.