Circle geometry (or circle theorems) involves looking at angle properties when lines and shapes are drawn in circles. Once you can identify and label parts of a circle, it is possible to add lines, triangles and quadrilaterals inside a circle. Circle theorems can help find missing angles.
Circle properties
You should be able to label parts of a circle.
NAME
DESCRIPTION
ILLUSTRATION
Radius
A line from the centre to the edge of the circle.
Diameter
A line across the circle, going through the centre.
Circumference
The perimeter of the circle.
Arc
A section of the circumference.
Chord
A line segment joining two points on the circumference.
Sector
Part of a circle bound by an arc and two radii.
Segment
Part of a circle bound by a chord and an arc.
Circle theorems
There are eight circle theorems that you should learn.
THEOREM
DESCRIPTION
ILLUSTRATION
Angles in the same segment are equal.
∠CAD=∠CBD
The angle at the centre is twice the angle at the circumference.
∠COB=2×∠CAB
The angle in a semicircle is 90°.
∠ACB=90°
The radius and tangent are perpendicular.
∠CAP=90°
The perpendicular bisector of a chord goes through the centre.
AB⊥OC
Opposite angles in a cyclic quadrilateral add up to 180°.
∠a+∠b=180°
∠c+∠d=180°
Two tangents from the same point are the same length.
AC=BC
The alternate segment theorem.
The angle at the tangent is equal to the angle in the alternate segment.
Example 1
In the circle shown, ∠AOD=120°. Find ∠ABD, ∠DBC and ∠DOC.
∠AODis twice the angle at the circumference ∠ABD.
∠AOD∠ABD∠ABD=2×∠ABD=2120=60°
∠ABC=90°as it is the angle in a semicircle. Therefore ∠ABD and ∠DBC add up to 90°.
The alternate segment theorum states that the angle at the tangent is the same as the angle in the alternate segment.
What is an arc?
An arc is a section of the circumference.
What is a cyclic quadrilateral?
A cyclic quadrilateral is a quadrilateral that is inside a circle, where the vertices of the quadrilateral touch the circumference of the circle. Opposite angles in a cyclic quadrilateral add up to 180 degrees.