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

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 always a on arc
(Called the Angles Subtended by Same Arc Theorem)

And 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

Try it here (not always exact due to rounding):


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.)

Play with it here:


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°).

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°


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

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!

Drawing a Circle From 2 Opposite Points

When we know two opposite points on a circle we can draw that circle.

Put some pins or nails on those points and use a builder's square like this:

  finding as circles 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 line just touches a circle at one point.

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


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