Angles and Circles


		Let me start by defining some terms that you need to know about circles. Most of these you will probably already know, but I think it's worth repeating them, don't you?

Every point on the edge of the circle is the same distance from the central point, and this distance is called the radius (r).
Any straight line that goes from one point on the edge of the circle, through the centre, to the point on the opposite edge of the circle, is called a diameter (d). The diameter is always twice the radius (d = 2r).

Any line that goes from one point on the edge of the circle to another point on the edge, but that doesn't go through the centre, is called a chord.

The distance all the way round the edge of a circle (its perimeter) is called its circumference.
The area enclosed within two radii (the plural of radius) of the circle and its edge (a "piece of the pie" with the pointed bit at the centre of the circle) is called a sector.

Any part of the circle's circumference is called an arc.
Any line that touches a circle, but doesn't actually cut across it, is called a tangent.

If you would like a short test on these technical terms, click on the button below:

Some theorems

If you draw a line from the centre of a circle to the middle point of a chord, then that line is perpendicular to the chord (i.e. at right angles to it). You can say this theorem the other way round (called a corollary), i.e. that a line drawn at 90^o to a chord and passing through the centre of the circle must touch the chord at its mid-point.

In that last diagram, I have used the convention that the centre of the circle is labelled with the letter O. This is quite common in maths questions, but if O is the centre of the circle, the question should at least say so!

You can probably see that the theorem must be true, as the line from the centre to the chord's mid-point is a line of symmetry. However, if you want a slightly more rigorous proof, then click here.

Imagine a triangle that is inscribed in a circle. This means that all the vertices (corners) of the triangle lie exactly on the edge of the circle. Let's also imagine that the longest side of the triangle is a diameter of the circle (i.e. it goes through the centre of the circle, O). In fact, you don't have to tax your imaginations too much, as all this is shown in the diagram on the left.

In this case, the angle of the triangle opposite the diameter (angle B in the diagram above) must be 90^o. We say that the diameter subtends a right angle at the circumference. If you would like a proof of this, then click here.

Here's the last theorem that we will meet for the moment.
If you draw a tangent to a circle and a radius from the centre of the circle to the point where the tangent touches it, then the radius and the tangent are perpendiular.

I'm afraid I can't offer any mathematically rigorous proof for this theorem. Sorry! All I can say is that the radius is line of symmetry for the circle, so the two angles that it forms with the tangent must be the same - they must both be 90^o.


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