Circle Theorems for GCSE/iGCSE

Circle Terminology

circleterminology

A circle is a 2-dimensional shape made by drawing a curve that is always the same distance from a fixed point.
A radius is a straight line from the centre of a circle to the circumference and is the same length all the way round the circle.
A chord is a straight line segment whose endpoints both lie on the circle.
A diameter is a straight line (chord) which goes through the centre of the circle. The length of a diameter is double that of a radius.
A secant is a line which cuts through a circle at two distinct points.
A tangent is a line which touches a circle once on its circumference.

Circle Theorems you need to know:

1. The angle at the circumference in a semicircle is a right angle.

right-angle

(Note: The line in the middle must be a diameter.)

In this diagram we can see that the green square is a right angle or 90 degrees.

Proof

Drawing a line from the centre of the circle to the opposite angle would give two isosceles triangles because all radii are equal.
Because they are isosceles, the size of the base angles are equal.
By exterior angle theorem, the angle’s size must be the sum of the other two interior angles.

2. The angle at the circumference is half as big as the angle in the centre.

circcentdouble

In this diagram we can see that the angle at the centre is twice that at the circumference as it is 2 lots of x.

Proof

When the arc is a semicircle, like above, the angle at the centre is a straight angle of 180 degrees and the angle at the circumference is a right angle (which is half).

In other words, if you look at theorem one, you will see that the angle at the circumference of the circle is 90 degrees and the angle at the diameter is 180 degrees as a diameter is two radii.

3.Angles in the same segment are equal.

samesegmentras

In this diagram one can see that both the red angles are the same size and both of the green angles are the same size.

4. The angle between a tangent and radius is always 90°.

tangentcircle90

5. In a cyclic quadrilateral, opposite angles add up to 180°.

opposite-angles

When there are four points, we can always draw a circle through any three of them (as long as they are not collinear), but the circle will only pass through the fourth point in special cases. A cyclic quadrilateral is a quadrilateral where all the corners (vertices) lie on a circle.

In this diagram the two orange angles add up to 180 degrees and the two green angles add up to 180 degrees. It may also be helpful to know that all the interior angles add up to 360 degrees.

6. Tangents which meet at the same point are of equal lengths.

twotangent

You can see a demonstration of circle theorem number five here as well. It is important to note that a line from the centre to the point where the two tangents meet would give you two identical triangles. You may also be able to use sin, cosine, or right angled triangle theorems to work out other angles or lengths from these triangles.

7. Alternate Segment Theorem – the angle between a tangent and a chord is the same as the angle in the alternate segment.

alternate-segment

This is probably one of the hardest circle theorems to spot. The easiest way to find one is by looking for a triangle with one of its corners (vertices) touching a tangent.