# Angles in a circle

**Right angles in a semicircle**

The angle in a semi-circle is always 90º.

**Calculating the size of unknown angles**

The labelled angles are calculated in the following diagram as follows:

a = 37º (angle at the base of an isosceles)

*b* = 90º – 37º

*b* = 53º (angle in a semi-circle)

*c* = 53º (angle at base of an isosceles triangle)

*d* = 180º – 2 x 53°

*d* = 74º (angle in a triangle)

*e* = 180º – 74º

*e* = 106º (angles on a line)

**The angle at the centre**

The angle at the centre of a circle is twice the angle at the circumference.

**Calculating the size of unknown angles**

Calculate the size of the labelled angles in the following diagram:

a = 29º (angle at the base of an isosceles)

*b* = 180º – (2 x 29º)

*b* = 122º (angles in a triangle)

*c* = 122º ÷ 2

*c* = 61º (angle at the centre is twice angle on the circumference)

*d* = 180º – (29° + 29° + 41° + 61°)

*d* = 20º – (angles in a triangle)

**Angles in a cyclic quadrilateral**

Opposite angles in a cyclic quadrilateral add up to 180° and when two angles add up to 180º, they are called supplementary angles. For example:

The centre of the circle should be marked as O and the angles labelled as ABC and SDC *x* and *y*. The two angles in the centre are then marked as 2*x* and 2*y.*

2*x* + 2*y = *360º

2(*x* + *y) = *360º

*x* + *y = 180*º

Point to note: The angle at the centre of a circle is twice the angle at the circumference.