Geometry & measures · Circles, arcs & sectors
Chapter 1 · 4
The idea
Circle theorems
The theorems about angles in a circle — at the centre, in a semicircle, in the same segment, and in a cyclic quadrilateral — why each is true, and the exact wording examiners require as a reason.
A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.
In this lesson — start anywhere
Geometry & measures · Circles, arcs & sectors
Circle theorems
The theorems about angles in a circle — at the centre, in a semicircle, in the same segment, and in a cyclic quadrilateral — why each is true, and the exact wording examiners require as a reason.
Why it works
One fact underneath: all radii are equal
Every circle theorem below comes from one fact: all radii are equal, so any triangle with two radii as sides is isosceles.Centre = twice the circumference
The angle at the centre is twice the angle at the circumference (on the same arc). Draw the radius from the centre to the point on the circumference. It splits the figure into two isosceles triangles; in each, the exterior angle at is twice the base angle at (exterior angle = sum of the two equal interior ones). Adding the two halves gives centre circumference.Keep reading — free
The rest of the explanation, plus 2 worked examples you step through move by move.
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