📐 Geometry
🎡Circle Theorems
Three angle theorems about central angles, inscribed angles and diameters — the rules a circle quietly enforces.
Why are wheels always round? Because every point of a circle sits at the same distance from the center — so the ride never bumps. Inside this "absolutely fair" shape hide several beautiful laws about angles.
Get the names right first
The center is ; the segment joining the center to a point on the circle is the radius, and its doubled cousin through the center is the diameter. You have met these parts in circles. Today's stars are two kinds of angles — both ordinary angles, measured the usual way:
- A central angle: vertex at the center, arms touching two points on the circle;
- An inscribed angle: vertex on the circle itself, arms along two chords.
The stretch of circumference caught between the arms is an arc — and the partnership between arcs and angles is today's main plot (it plays bigger roles in arcs and sectors).
Diameter d
d = 6
Circumference C = 2πr
C = 18.85
Area A = πr²
A = 28.27
C = 2πr = 2π×3 ≈ 18.85 · A = πr² = π×3² ≈ 28.27
Drag the radius slider and watch diameter, circumference and area follow — the diameter is always twice the radius. Warmed up? On to the three theorems.
The inscribed angle theorem: the half relationship
Look at the same arc once from the center, then from any point on the circle: the inscribed angle is always half the central angle standing on the same arc. From the center the arc is close, so the view is widest; step back to the rim and the very same arc shrinks to half its apparent size. A central angle of 80° comes with an inscribed angle of 40°; 90° comes with 45°.
Inscribed angles on the same arc are equal
Two students stand at different spots on the circle (same side) and look at one shared arc: the inscribed angles they measure are equal — each is half of the very same central angle, and one half equals the other. With a central angle of 60°, wherever you stand on the arc's side of the circle, the inscribed angle reads 30°. "Same arc" acts as a circle's equality certificate: change your position, not your angle.
An angle in a semicircle is 90°
The most special arc is the semicircle — its endpoints joined by a diameter. The central angle on a diameter is the straight angle 180°, so from any point on the circle, the inscribed angle facing that diameter is always . In other words: a triangle inscribed in a circle with one side a diameter must be a right triangle, with the diameter as its hypotenuse — a neat handshake with the Pythagorean theorem. Need a right angle without a protractor? Draw a circle, pull a diameter, mark any point on the rim — three moves and done.
All three theorems in one go
An arc faces a central angle of 100°, and , are two points on the same side of the circle: — each is half the central angle, hence equal to each other. Swap the arc for a semicircle (central angle 180°) and the inscribed angle becomes 90°. The three theorems are one family: all of them grow out of "inscribed angle = central angle ÷ 2".
Check yourself
Quick quiz
1. An arc faces a central angle of 100°. What is its inscribed angle?
2. AB is a diameter and C is any point on the circle. What is ∠ACB?
3. C and D sit on the same side of the circle, both looking at arc AB. How do ∠ACB and ∠ADB relate?