Circle Theorems


circle1

The red line in the second diagram is called a chord. It divides the circle into a major segment and a minor segment.

circle2

The diagrams show that:
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, a = 2b.
b) Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
c) Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So in the third diagram, c is a right angle.
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.

Area of Sector and Arc Length

circle3

If the radius of the circle is r,
Area of sector = pr² × A/360
Arc length = 2pr × A/360

In other words, area of sector = area of circle × A/360
arc length = circumference of circle × A/360