# Understanding Circles

• A circle is a set of all points in a plane that are at a given distance from a single point called the centre.
• Recognise the different parts of a circle: the radius is a line segment connecting the centre with any point on the circle’s edge, the diameter is a line segment passing through the centre and touching two points on the edge, and the circumference is the edge of a circle.
• The tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of tangency.
• A chord of a circle is a straight line joining two points on its circumference.
• An arc is a portion of the circumference of a circle.
• Understand a sector as a portion of a circle, enclosed by two radii and an arc.
• A segment is a part of a circle defined by a chord and the associated arc.

# Circle Properties and Theorems

• The diameter is twice the length of the radius.
• The circumference of a circle can be calculated using the formula C = 2πr or C = πd.
• The area of a circle can be calculated using the formula A = πr².
• Any angle at the centre of a circle standing on a given arc is twice the size of the angle at the circumference standing on the same arc.
• The perpendicular from the centre of a circle to a chord bisects the chord.
• Tangents drawn from a common point outside a circle are equal in length.
• The angle between the radius and a tangent to a circle is always 90°.
• The opposite angles of a cyclic quadrilateral (a four-sided figure whose vertices all lie on a circle) add up to 180°.

# Solving Circle Problems

## Recognising the Problem

• Identify problems that involve circles based on the presence of radius, diameter, circumference, area, sectors, segments, or arcs.
• Pay attention to words that indicate problems about tangents, chords, or angles related to circles.

## Applying Circle Formulas

• Use the circle formulas for circumference (C = 2πr or C = πd) and for area (A = πr²) to solve problems.
• Calculate arc length and sector area by first determining the fraction of the whole circle represented by the given angle (angle/360), then multiplying this fraction by the whole circle’s circumference or area.

## Utilising Circle Theorems

• Understand the significance of circle theorems and apply them to solve problems involving angles, chords, tangents, and cyclic quadrilaterals.