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.

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

Play with it here:

When you move point "B", what happens to the angle?

Inscribed Angle Theorems

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

inscribed angle a on circumference, 2a at center
(Called the Angle at the Center Theorem)

And (keeping the end points fixed) ...

... the angle is always the same,
no matter where it is on the same arc between end points:

inscribed angle alwyas a on 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)

inscribed angle 62 on circumference

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

Example: What is the size of Angle CBX?

inscribed angle example

Angle ADB = 32° also equals Angle ACB.

And Angle ACB also 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 (Thales' Theorem)

An angle inscribed across a circle's diameter is always a right angle:

angle inscribed across diameter is 90 degrees
(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)

angle semicircle 90 degrees and 180 at center

 

Another Good Reason Why It Works

angle semicircle rectangle

angle semicircle rectangle

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°).

 

angle semicircle always 90 on circumference
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?

inscribed angle example

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°

 

Finding a Circle's Center

finding as circles center

We can use this idea to find a circle's center:

Where the diameters cross is the center!

 

Cyclic Quadrilateral

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

quadrilateral cyclic

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

  • a + c = 180°
  • b + d = 180°
quadrilateral cyclic a and c add to 180

Example: What is the size of Angle WXY?

inscribed angle example

Opposite angles of a cyclic quadrilateral add to 180°

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

 

90 degrees between radius and tangent

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.