Circle Theorems

Some interesting things about angles and circles.

Inscribed Angle

First off, a definition:

Inscribed Angle: an angle made from points sitting on the circle's circumference.


A and C are "end points"
B is the "apex point"

Inscribed Angle Theorems

An inscribed angle is half of the central angle 2a°


(Called the Angle at the Center Theorem)

And (keeping the endpoints fixed) ...

... the angle is always the same, no matter where it is on the circumference:


Angle a° is the same.
(Called the Angles Subtended by Same Arc Theorem)

 

Example: What is the size of Angle POQ? (O is circle's center)

Angle POQ = 2 × Angle PRQ = 2 × 62° = 124°

Example: What is the size of Angle CBX?

Angle ADB = 32° equals Angle ACB.

And Angle ACB equals Angle XCB.

So in triangle BXC we know Angle BXC = 85°, and Angle XCB = 32°

Now use angles of a triangle add to 180° :

Angle CBX + Angle BXC + Angle XCB = 180°
Angle CBX + 85° + 32° = 180°
Angle CBX = 63°

Angle in a Semicircle

An angle inscribed in a semicircle is always a right angle:


(The end points are either end of a circle's diameter,
the apex point can be anywhere on the circumference.)

Why? Because:

The inscribed angle 90° is half of the central angle 180°

(Using "Angle at the Center Theorem" above)


 

Another Good Reason Why It Works

We could also rotate the shape around 180° to make a rectangle!

It is a rectangle, because all sides are parallel, and both diagonals are equal.

And so its internal angles are all right angles (90°).

 


So there we go! No matter where that angle is
on the circumference, it is always 90°

Example: What is the size of Angle BAC?

The Angle in the Semicircle Theorem tells us that Angle ACB = 90°

Now use angles of a triangle add to 180° to find Angle BAC:

Angle BAC + 55° + 90° = 180°
Angle BAC = 35°

 

Cyclic Quadrilateral

A "Cyclic" Quadrilateral has every vertex on a circle's circumference:

A Cyclic Quadrilateral's opposite angles add to 180°:

  • a + c = 180°
  • b + d = 180°

Example: What is the size of Angle WXY?

Opposite angles of a cyclic quadrilateral add to 180°

Angle WZY + Angle WXY = 180°
69° + Angle WXY = 180°
Angle WXY = 111°

 

Tangent Angle

A tangent is a line that just touches a circle at one point.

It always forms a right angle with the circle's radius.