It’s not exactly a geometric object, but it is the **length** of a geometric object.

Draw a circle of radius 1 and two rays from the center of the circle. Two rays form an angle. The “sine” of that angle is the length of the shortest possible line segment that connects from one of the rays to the point where the other ray intersects the circle.

You could also say “the length of the line segment that is perpendicular to one of the rays and intersects the point where the other ray intersects the circle”.

For example, if the angle formed is 30 degrees, the line segment has length 0.5, hence the “sine” of 30 degrees is 0.5 .

P.S.

I must admit that Euclid, Pythagoras, et al, would not have looked at it this way. Here in the 21st century, we tend to think of fractions as numbers, but they would have thought of a fraction as a ratio. When we talk about the number ½, we think of a point on the number line which is halfway between zero and one, represented by the decimal 0.5 . But they would have thought of it as two line segments, one of which is exactly twice as long at the other. Hence, they would not have started by saying the circle has radius 1, they would have said a *radius* is a line segment connecting the center of the circle to the edge of the circle. So, from their perspective…

Construct a circle (of any size). The center of the circle is a point, which we call “*c*”. Construct a ray starting at *c*. The point where the ray intersects the circle is “*a*”. Line segment ac is known as a radius. Construct another ray from *c*. From point *a*, construct a line segment which is perpendicular to the second ray and intersects the second ray at a point we will call “b”. This gives us line segment *ab*. The two rays form an angle. The “sine” of that angle is the ratio of *ab* to *ac*.

Again, take for example the angle we call 30 degrees. They would have said that the measure of the angle is the ratio of that angle to a right angle, which is 1:3, and sine of that angle is the ratio of ab to ac, which is 1:2. From their perspective, the measure of an angle is a ratio and the sine of that angle is also a ratio. The connection between the ratios is determined by the geometric construction I described above.

P.P.S. This perspective about ratios explains why it was so hard for them to accept the existence of irrational numbers, which they would have called “incommensurable distances”. The believed that, given any two line segments, either one of the line segments could evenly divide the other or there had to be a third segment which would evenly divide both of the first two. Given ab which is larger than cd, either cd divides ab n times (in which case the ratio is n:1) or there must exist ef which divides ab n times and divides cd m times (in which case the ratio is n:m). It was embarrassing to them that when you bisect a right angle, you get an angle whose sine cannot be written as a ratio of whole numbers. In that example, when you follow my construction above, you get ab and ac but there is no way to write their ratio with whole numbers. We can narrow it down, e.g. the ratio of ab to ac must be less than 5:7 and greater than 9:13, but every ratio of whole numbers is either less than or greater than, never equal to, the sine of 45 degrees, sqrt(2)/2.