Circle Properties

The Basics

Circumference – the perimeter of a circle

Diameter – the distance across a circle, passing through the centre

Radius – the distance from the centre of a circle to the circumference, plural of radius is radii

Tangent – a straight that touches the circumference of the circle at a single point

Chord – a straight line joining two points on the circumference of the circle

Arc – part of the circumference of a circle

Segment – the area inside a circle bounded by an arc and a chord, a chord divides a circle into two segments

Sector – the area inside a circle bounded by an arc and two radii

Semicircle – the area bounded by a diameter and an arc

Circle Geometry

There are 9 rules to remember covering circle geometry. Make sure you are familiar with them and know how to use them.

Angle in a semicircle = 90ْ

A triangle drawn from the two ends of a diameter makes an angle of 90ْ at the circumference

Tangent and radius meet at 90ْ

If a tangent and radius meet at the same point on the circumference of a circle, the angle made between the radius and the tangent is 90ْ

 Isosceles triangles formed by two radii

 Triangles where two sides are formed by radii of a circle are isosceles triangles. If two sides of a triangle are both radii of the same circle, the two sides must be the same length.

Chord bisector is a diameter

The perpendicular bisector of a chord is a diameter. If a diameter cuts a chord in half, it meets the chord at a 90ْ angle. The perpendicular bisector of a chord passes through the centre of the circle and so is a diameter.

 Angles in the same segment are equal

All triangles drawn from a chord into the same segment form the same angle where they touch the circumference. Two triangles drawn from a chord into opposite segments of a circle form angles that sum to 180ْ at the circumference.

 Angle at the centre is twice the angle at the circumference

If you draw two triangles in a circle from the same chord, one to the centre of the circle and one to the circumference of the circle, the angle formed at the centre of the circle is double the angle formed at the circumference.

Opposite angles of a cyclic quadrilateral add up to 180ْ

A cyclic quadrilateral is a four-sided shape with each corner touching the circumference of a circle. Diagonally opposite angles of the cyclic quadrilateral add up to 180ْ

 Equality of tangents from a point

 If you draw two tangents from any given point outside a circle, the tangents are of equal length from the point to where they meet the circle. If you draw the two tangents from a point, and then also draw two radii to form two triangles, the triangles are a pair of congruent right-angled triangles.

Angle in opposite segment is equal

 If you draw a tangent and a chord that meet at a point on the circumference of a circle, the angle between the chord and the tangent is equal to the angle in the opposite segment.

Download

Diagrams showing all of the above circle properties and the 9 rules of circle geometry. (Please note, this is a zipped file. If you have trouble downloading or unzipping the file please email me.)