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Revision:Circle Theorems
TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Circle Theorems
The red line in the above diagram is called a chord, and separates the circle into two segments, one minor (smaller) and one major (larger). (If the two segments are the same size, then the chord passes through the centre and is called a diameter. The radius is the distance from the centre of the circle to any point on the edge, i.e. half the length of the diameter.) The blue lines separate the circle into two sectors, again one minor and one major. The line that the circle is made up of is often called the circumference; any section of this circumference is referred to as an arc. Any arc defined by a minor segment or sector will be a minor arc, and vice-versa.
a) The angle formed at the centre of the circle by lines originating from two points on the circle's circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. in the first diagram, 
b) Angles on a chord are always equal.
c) Angles in a semicircle are always right angles.
d) A tangent to a circle forms a right angle with the circle's radius, at the point of contact of the tangent (a tangent to a circle is a line that touches the circumference at one point only).
e) The final diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.
A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.
If the radius of the circle is r,
Area of sector = 
Arc length = 
In other words, area of sector = area of circle 
arc length = circumference of circle
