Circle Theorems
(KS3, Year 8)

Circles have properties relating to angles and lines. There are several circle theorems that apply to all circles.

A Tangent and a Radius Meet at 90°

The tangent makes 90° with the radius which it meets at the point at which it touches. circle_theorem_tangent_radius_90_degrees

Two Radii Form an Isosceles Triangle

Two radii form the two equal sides of an isosceles triangle. Note: Radii is the plural of radius. circle_theorem_two_radii_make_isosceles_triangle

The Perpendicular Bisector of a Chord Passes Through the Center of the Circle

If a line cuts through a chord of the circle, such that it crosses it at 90° and cuts it in half, that line passes through the centre of the circle.

circle theorem perpendicular bisector of a chord passes through center

The Angle at the Center of a Circle Is Twice the Angle at the Circumference

The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference. circle theorem angle at center twice angle at circumference

The Angle in a Semicircle Is 90°

A triangle drawn from two ends of a diameter makes 90° at the circumference. circle theorem angle in semicircle is 90 degrees

Angles in the Same Segment Are Equal

All triangles drawn from a chord make the same angle at the circumference. circle theorem angles in same segment are equal

Opposite Angles in a Cyclic Quadrilateral Add Up to 180°

A cyclic quadrilateral is a 4 sided shape where each corner touches the circle. Both pairs of opposite angles add up to 180°.

circle theorem opposite angles in a cyclic quadrilateral add up to 180 degrees

Tangents from the Same Point Are the Same Length

Two tangents drawn from the same point outside of the circle are the same length and form two congruent right triangles.

circle theorem tangents from same point are equal

The Angle Between a Tangent and a Chord is Equal to the Angle in the Alternate Segment

The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. circle theorem angle in alternate segment is equal

What Is a Circle?

A circle is a shape containing a set of points that are all the same distance from a given point, its center.

Parts of a Circle

To understand the circle theorems, it is important to know the parts of a circle. parts_of_a_circle_explained

Useful Definitions

Here are some useful definitions of some words used to explain the circle theorems.
  • Perpendicular: means at right angles (90°) to.
  • Bisector: to bisect means to divide into two equal parts.
  • Subtended angle: the angle subtended by an arc is the angle between the lines drawn from the ends of the arc to the same point.
  • Semicircle: half a circle.
  • Cyclic quadrilateral: a four sided shape where each of the 4 corners lie on the circumference of a circle.
  • Alternate segment: the alternate segment of a chord is formed by drawing lines from each end of the chord to a point on the circumference of the circle.
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    This page was written by Stephen Clarke.