The ancient Greeks did it, and you can too. Find out how some easily observable facts allow you to measure the approximate distance from the Earth to the Moon.
One of the hardest parts of calculating distances in space is the difficulty finding reference points. The size or distance of objects on Earth can be hard to estimate, but they occupy a landscape which can be measured, providing a jumping-off point. The moon gives up a few clues — it's clearly closer than the Sun or the stars, but it's still drifting in a nothingness that's hard to measure.
The distance to the moon was measured, or at least approximated, over 2000 years ago, by our old friends, the Greeks. They'd already figured out the circumference and consequently the diameter of the earth, providing the one absolute number on which to base the rest. After that, it's geometry.
Many people have held up a round object and let it block the Sun. Most of the time, it's not an exact fit. A slice of Sun peeks through, or a little of the surrounding area is blocked out. When a round object is held up in front of the Sun, it creates a cone of darkness that tapers down to one point. At that one point, the object blocks out all of the Sun, and nothing else. That point, on Earth, is 108 times the diameter of the object. A beach ball will create a shadow 108 beach balls long, which at the farthest point will block out the Sun completely. A penny will create a shadow 108 pennies long. The Earth will create a shadow 108 Earth diameters long.
The Moon passes within that shadow during a lunar eclipse. So no matter how big or small the Moon is, it had to pass within 108 Earth diameters of the Earth. In fact, during lunar eclipses, it was observed that the Moon was imperfectly blocked by the shadow of the Earth. The shadow was roughly 2.5 times the width of the Moon.
But is it a big, far Moon, or a small, close Moon? This would have been impossible to decide if it weren't for a happy coincidence. The Moon itself is a size and distance that blocks out the Sun. Like the beach ball and the penny, it creates its own shadow, and that shadow terminates on Earth. More importantly, that shadow ends in the same angle that the shadow of the Earth does, making them different-sized versions of the same triangle.
The triangles work out like this. The largest is one Earth diameter wide at the base (8,000 miles) and 108 Earth diameters tall (864,000 miles). The smallest is one Moon diameter wide and one Moon orbit tall. The medium sized one is 2.5 Moon diameters wide and, since the triangles are proportionate, 2.5 Moon orbits tall. Add the height of the medium-sized one to the small one and you get 3.5 Moon orbits, which is the height of the largest triangle.
In other words, the distance to the Moon is 864,000 divided by 3.5, or around 247,000 miles. According to Universe Today, the distance to the Moon is 239,000 miles, proving once again that the Greeks were smart.
(Note of Caution: Don't make fun of my MS paint skills.)