Lecture · 14 min · intermediate
Circles
Inscribed angles, tangents, and π as a ratio
Euclid, Elements · Book III
is not . It is the ratio of circumference to diameter, the same for every circle — a theorem, not a definition of a number you type into a calculator. The rest of circle geometry is about how angles sitting on the same arc compare, and how a tangent meets a radius.
On the board today
- —Relate a central angle to an inscribed angle on the same arc.
- —Use the tangent–radius right angle.
- —Treat as a ratio, independent of the circle you measured.
An angle at the circumference is half the angle at the centre, same arc .
Corollary: an angle inscribed in a semicircle is a right angle (Thales). A tangent is perpendicular to the radius at the point of contact — that is the cleanest definition of “just touching.” Two tangents from an external point are equal in length.
Worked example · Angle in a semicircle
Diameter of a circle, point on the circle. Why is ?
- 1.The central angle of arc is (a diameter).
- 2.The inscribed angle is half of that: .
Thales' theorem is the inscribed-angle theorem on a semicircle.
Pause the lecture
If a central angle is , what is an inscribed angle on the same arc?
Common mistake. Calling every angle that “looks like it sits on the circle” inscribed. The vertex must lie on the circle, and the sides must be chords (or a chord and a tangent, for the alternate segment theorem).