Central Angle in a Circle

We will mark the center of the circle with the letter $A$.

We note that a central angle is an angle whose vertex is the center of the circle
and its sides are the radii of the circle.
Therefore, if we draw two radii, an angle will be formed.
The angle that will be created will be a central angle in the circle.

Now that we already know what a central angle in a circle is and can easily identify it,
we must learn about some important theorems and characteristics of a central angle in a circle.
Shall we begin?

When can we determine that two central angles in a circle are equal?
In two cases:

• If the arcs on which the angles are based are equal, then we can determine that the central angles are equal.

Let's see this in the figure:

If $BC=DE$
Then $∢A1=∢A2$

If the arcs in front of the central angles are equal, then the central angles are equal.

In the same way, the theorem works in reverse.
If the central angles are equal, the arcs in front of them are also equal.

• If the chords opposite the central angles are equal, then we can determine that the central angles are equal.

Let's see this in the figure:

If $BC=DE$
then $∢A1=∢A2$

If the chords in front of the central angles are equal, then the central angles are equal.
In the same way, the theorem works in reverse.
If the central angles are equal, the chords in front of them are also equal.

Now we will learn the relationship between a central angle in a circle and an inscribed angle in a circle.
Remember that an inscribed angle in a circle is an angle whose vertex is on the circumference of the circle and whose ends are chords in a circle.
Like here:

In a circle, the central angle will be twice the inscribed angle that spans the same arc.
Or, the inscribed angle will be equal to half the central angle that spans the same arc.

That is:
If in the circle we identify a central angle and an inscribed angle that lean on the same arc, we can say that as they lean on the same arc, the inscribed angle will be equal to half of the central angle or alternatively, the central angle will be double the inscribed angle.
And we will see it in the illustration:

Before us, there is a circle.

We can identify that the angle $A$ is a central angle, which comes from the vertex of the circle and its radii at the ends, while an angle $B$ is an inscribed angle - a vertex is on the circle and its ends are chords.
We can also see that these two angles are subtended - they are on the same arc - $CD$
Therefore, we can conclude that the inscribed angle $B$ is equal to half of the central angle $A$ or alternatively, the angle $A$ is equal to twice the angle $B$.

$∢A=α$
$∢B=1/2 α$

or 

$∢A=2α$
$∢B=α$

Wonderful!

Now we understand the relationship between an inscribed angle and a central angle. It is worth knowing more different representations of inscribed and central angles that are on the same arc:

If in a circle we are given that the inscribed angle is $5$ degrees and we are asked what the value of the marked central angle is,
we can see that they lean on the same arc and therefore determine that the central angle is $2$ times greater than the inscribed angle that leans on the same arc.
It is deduced
that the marked central angle is equal to $10$.
$5\times2=10$

If you are interested in this article, you might also be interested in the following articles:

• The center of the circle
• Circle
• Radius
• Diameter
• Pi
• The circumference perimeter
• Circular area
• Arcs in a circle
• Chords in a circle
• Perpendicular to the chord from the center of the circle
• Inscribed angle in a circle
• Distance from the chord to the center of the circle

In the Tutorela blog, you will find a variety of articles about mathematics.