Central Angle

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Central angle

Here you will learn about central angles of circles, inscribed angles of circles, and the central angle theorem. You will learn how to problem solve, apply the central angle theorem to solve problems, and apply the central angle theorem to solve more difficult problems.

Students first learn about central angles in geometry and expand their knowledge as they progress through high school level math classes.

What is a central angle?

The central angle of a circle is an angle that has its vertex at the center of the circle and the lines creating the angle are two radii of the circle.

The length of the arc is dependent on the size of the central angle.

For example, in the circle below, angle $BAC$ is a central angle because the vertex is the center of the circle with $AB$ and $AC$ being two radii.

The measurement of angle $BAC$ is $90^{\circ}$ which means that the arc length is $\cfrac{90}{360}$ of the circumference.

An inscribed angle is an angle that has its vertex on the edge of the circle, and the sides are chords.

Angle $BDC$ is an inscribed angle of the circle that intercepts arc $BC.$

The inscribed angle is half the measure of the arc it intercepts. Continuing with the same example, the angle $BDC=45^{\circ}.$

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[FREE] Angles Check for Understanding Quiz (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

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Central angle theorem

The central angle theorem (sometimes called the inscribed angle theorem) states that:

“The central angle of a circle is twice the measure of the
inscribed angle intercepting (or substending) the same arc.”

In the diagram below, angle $DAB$ is $\cfrac{1}{2}$ the measure of angle $DCB.$

Notice how both angles intercept arc $DB.$ Arc $DB$ is equal to the measure of angle $DCB.$

Before applying this theorem to problem solve, let’s look at how this relationship is always true.

The equilateral triangle $ABC$ is inscribed in circle $D.$ Each of the angles of the triangle are $60^{\circ}.$

Split the equilateral triangle into three congruent isosceles triangles by connecting each vertex to the center of the circle. Now triangle $ADB =$ triangle $BDC =$ triangle $CDA.$

They are isosceles triangles because each of the congruent sides $AD, \, BD$ and $CD$ are the radii of circle $D$ and all radii within the same circle are congruent in length.

Notice, how the vertex angles of each of the isosceles triangles have a measure of $120^{\circ}$ and each interior angle of the original triangle $ABC$ is halved to $30^{\circ}.$

Angle $ABC$ is an inscribed angle that has a measure of $60^{\circ}.$ Angle $ADC$ is a central angle that has a measure of $120^{\circ}.$

The inscribed angle is $\cfrac{1}{2}$ the measure of the central angle, and both angles intercept the same arc, arc $AC.$

This relationship is always true within any circle.

What is a central angle?

Common Core State Standards

How does this relate to high school math?

High School Geometry – Circles: (HSG-C.A.2)
Identify and describe relationships among inscribed angles, radii, and chords.

Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

How to use central angles to solve problems

In order to use central angles to solve problems:

Identify the key parts of the circle.
Apply the central angle theorem to problem solve.
State the answer.

Central angle examples

Example 1: standard diagram inscribed angle

Use the diagram below to find the measure of angle $BAD$ in circle $C.$

Identify the key parts of the circle.

$C$ is the center of the circle.

Angle $BDC$ is the central angle of the circle because the vertex is at the center and the sides are the radii of the circle.

Angle $BAD$ is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.

2Apply the central angle theorem to problem solve.

The central angle, angle $BDC=150^{\circ}$ which means the angle it intercepts is also $150^{\circ}.$ Arc $BD=150^{\circ}$

The inscribed angle, angle $BAD$ intercepts arc $BD.$

The central angle theorem states that the inscribed angle is $\cfrac{1}{2}$ the measure of the central angle that intercepts the same arc.

So, angle $BAD=\cfrac{1}{2}\times{150}=75^{\circ}.$

3State the answer.

Angle $BAD=75^{\circ}$

Example 2: standard diagram central angle

Use the diagram below to find the measure of the central angle, angle $MLO$ in circle $L.$

Identify the key parts of the circle.

$L$ is the center of the circle.

Angle $MNO$ is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Angle $MLO$ is the central angle because the vertex is the center of the circle and the sides are the radii.

The measure of the inscribed angle is $52^{\circ}.$

Apply the central angle theorem to problem solve.

The central angle is twice the measure of the inscribed angle that intercepts the same arc.

Since the measure of the inscribed angle is $52^{\circ},$ using the central angle theorem, angle $OLM$ is $2\times{52}=104^{\circ}.$

State the answer.

Angle $MLO=104^{\circ}$

Example 3: two inscribed angles

Use the diagram below to find the value of the central angle $MHN.$

Identify the key parts of the circle.

$H$ is the center of the circle.

Angle $MIN$ and angle $MGN$ are inscribed angles because their vertices are on the edge of the circle and their sides are chords. They are also the same size.

Step-by-step guide: Subtended Angles in the Same Segment

Angle $MIN =$ angle $MGN,$ since both inscribed angles subtend the same arc so:

\begin{aligned}3x+13&=6x-5 \\\\ 3x+13+5&=6x-5+5 \\\\ 3x+18&=6x \\\\ 3x-3x+18&=6x-3x \\\\ 18&=3x \\\\ \cfrac{18}{3}&=\cfrac{3 x}{3} \\\\ 6&=x \end{aligned}

Since $x=6,$ substitute that value into either one of the expressions representing the inscribed angles to find the measure of the inscribed angles.

Angle $MIN=3(6)+18=36^{\circ}$

Now, Angle $MIN$ is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Angle $MHN$ is the central angle because the vertex is at the center of the circle and the sides are radii.

Apply the central angle theorem to problem solve.

Angle $MHN=2\times\text{angle}~MIN$

So, angle $MHN=2\times{36}=72^{\circ}.$

State the answer.

Angle $MHN=72^{\circ}$

Example 4: form and solve equations using the central angle

Use the diagram below to find the measure of arc $MN.$

Identify the key parts of the circle.

The center of the circle is $H.$

The central angle is angle $MHN$ because the vertex is at the center of the circle and the sides are radii.

The inscribed angle is angle $MIN$ because the vertex is on the edge of the circle and the sides are chords.

Apply the central angle theorem to problem solve.

The inscribed angle is $\cfrac{1}{2}$ the measure of the central angle that intercepts the same arc. So, in this case $\text{angle}~MIN=\cfrac{1}{2}\text{ (angle}~MHN).$

Substituting in the algebraic expressions, the equation is:

\begin{aligned}3x-4&=\cfrac{1}{2} \, (5x+9) \\\\ 2\times(3x-4)&=2\times\cfrac{1}{2} \, (5x+9) \\\\ 6x-8&=5x+9 \\\\ 6x-5x-8&=5x-5x+9 \\\\ x-8+8&=9+8 \\\\ x&=17 \end{aligned}

Substitute the value of $x$ back into the algebraic expression for $MHN$ to find the measure of arc $MN.$

Angle $MHN=5x+9=5\times{17}+9=94^{\circ}$

State the answer.

The measure of arc $MN=94^{\circ}$

Example 5: using triangles to find central angle

Use the diagram below to find the value of the central angle, $\theta$ (Angle $BCD$ ).

Identify the key parts of the circle.

The center of the circle is $C.$

The central angle is angle $BCD$ because the vertex is at the center of the circle and the sides are radii.

Angle $BAD$ is an inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Triangle $ABD$ is an inscribed triangle because the vertices of the triangle are on the edge of the circle.

To find the central angle, first you need to know the inscribed angle $BAD.$

As the inscribed angle is part of a triangle, use the angle fact “the sum of angles in a triangle is $180^{\circ}$ to find the angle.

$x=180-(39+104)=37^{\circ}$

Now, Angle $BAD$ is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Angle $BCD$ is the central angle because the vertex is at the center of the circle and the sides are radii.

Apply the central angle theorem to problem solve.

Angle $BAD$ is $\cfrac{1}{2}$ the measure of the central angle that intercepts the same arc, so:

$\text{angle}~BAD=\cfrac{1}{2}\times\text{angle}~BCD.$

Substitute in the value for angle $BAD$ .

\begin{aligned}37&=\cfrac{1}{2}\times\theta \\\\ 37\times{2}&=\theta \\\\ \theta&=74^{\circ} \end{aligned}

State the answer.

The central angle $BCD=74^{\circ}$

Example 6: right triangles

Use the diagram below to find the value of angle $XWV.$

Identify the key parts of the circle.

$Y$ is the center of the circle.

Angle $XWV$ is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Straight line, $XV,$ is the diameter of the circle because it is a chord that goes through the center of the circle.

Angle $XYV$ is the central angle which is also a straight angle $=180^{\circ}.$

Apply the central angle theorem to problem solve.

Angle $XYV=180^{\circ}$ because it is a straight angle and it also can be considered the diameter, which intercepts the semi-circle. The measure of arc $XV,$ is equal to $180^{\circ}.$

The inscribed angle $XWV$ intercepts the semi-circle so it is $\cfrac{1}{2}$ the measurement of the central angle that intercepts the same arc.

Angle $XWV=\cfrac{1}{2} \, (180)=90^{\circ}$

State the answer.

Angle $XWV=90^{\circ}$

Example 7: arcs and angles

Use the diagram below to find the measure of the inscribed angle $RTS$ and the central angle, $RQS$ of circle $Q.$

Identify the key parts of the circle.

$Q$ is the center of the circle.

Angle $RTS$ is the inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Angle $SQR$ is the central angle because the vertex is at the center of the circle and the sides are radii.

$ST$ is the diameter because it is a chord that goes through the center of the circle.

Apply the central angle theorem to problem solve.

Angle $RQT$ and the measure of semi-circle $SRT$ is $180^{\circ},$ so the measure of arc $SR=180-145=35^{\circ}.$

Arc $SR$ is intercepted by the central angle $SQR$ so the measurement of arc $SR$ is equal to the measurement of central angle $SQR.$

$\text{Arc}~SR=35^{\circ}=\text{angle}~SQR$

Inscribed angle $STR$ intercepts the same arc as central angle $SQR.$ So, inscribed angle $STR$ is $\cfrac{1}{2}$ the measurement of central angle $SQR.$

Angle $STR=\cfrac{1}{2} \times 35^{\circ}=17.5^{\circ}$

State the answer.

Angle $SQR=35^{\circ}$

Angle $STR=17.5^{\circ}$

Teaching tips for central angles

Have students discover the relationship between central angles and inscribed angles as well as the central angle theorem by doing hands-on constructions and measuring the relationships between the angles.

Instead of practice problem worksheets, have students game play to practice skills.

Easy mistakes to make

The inscribed angle is equal to the central angle
The inscribed angle is always double the size of the central angle, or vice versa, the central angle is always half the size of the inscribed angle.

In the example below, $\text{Angle}~BCD=2\times\text{angle}~BAD$ or
$\text{Angle}~BAD=\cfrac{1}{2}\times\text{angle}~BCD.$

Halving and doubling incorrectly
Similar to the previous misconception, the central angle is incorrectly doubled to calculate the inscribed angle whereas it should be halved.

Circle theorems
Chords of a circle
Tangent of a circle
Circle chord theorems

Central angle practice problems

144^{\circ}

36^{\circ}

18^{\circ}

108^{\circ}

Angle $ABC$ is a central angle because the vertex is at the center of a circle with the sides being radii.

Angle $ADC$ is an inscribed angle with vertex on the edge of the circle intercepting the same arc as angle $ABC.$

Using the central angle theorem, angle $ABC$ is double the measure of angle $ABC.$

72\times{2}=144^{\circ}

236^{\circ}

62^{\circ}

60^{\circ}

59^{\circ}

Angle $ADC$ is a central angle because the vertex is at the center of the circle and the sides are radii.

Angle $ABC$ is an inscribed angle because the vertex is on the edge of the circle and the sides are chords.

The central angle $ADC=118^{\circ}$ and the inscribed angle $ABC$ is half the measurement of angle $ADC$ because they both intercept the same arc.

118\times\cfrac{1}{2}=118\div{2}=59^{\circ}

31^{\circ}

236^{\circ}

124^{\circ}

298^{\circ}

Angle $BAD$ is an inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Angle $BED$ is a central angle because the vertex is at the center of the circle and the sides are radii.

Angle $BAD=62^{\circ}$ which means that angle $BED$ is double angle $BAD$ because the central angle is $2$ times the inscribed angle that intercepts the same arc.

Angle $BED=2\times{62}=144^{\circ}$

The sum of angles at a point is $360^{\circ}$ so the reflex angle at $E$ which is $\theta$ is equal to $360-144=236^{\circ}.$

48^{\circ}

132^{\circ}

24^{\circ}

66^{\circ}

Angle $SQR$ is a central angle because the vertex is at the center of the circle and the sides are radii.

Angle $STR$ is an inscribed angle because the vertex is on the edge of the circle with the sides being chords.

The minor arc measurement, arc $RT=132^{\circ}$ and the measurement of semi-circle $ST=180^{\circ}$ so arc $SR=180-132=48^{\circ}.$

The measure of the intercepted arc is equal to the measure of the central angle.

Arc $SR=48^{\circ}$ which means that central angle $SQR=48^{\circ}$ and inscribed angle $STR$ is half the angle $SQR.$

48\times\cfrac{1}{2}=48\div{2}=24^{\circ}

5^{\circ}

40^{\circ}

20^{\circ}

45^{\circ}

Angle $MJN$ and angle $MKN$ are both inscribed angles and intercept the same arc. So, they are both equal to each other.

The measure of the intercepted arc in this case is equal to the central angle measurement.

In order to find the measure of the central angle, angle $MHN,$ you must first solve for $x.$

\begin{aligned}2x+10&=5x-5 \\\\ 2x-2x+10&=5x-2x-5\\\\ 10&=3x-5 \\\\ 10+5&=3x-5+5 \\\\ 15&=3x \\\\ \cfrac{15}{3}&=\cfrac{3x}{3} \\\\ 5&=x \end{aligned}

Substitute the value of $x$ back into one of the algebraic expressions for the inscribed angle to find the measurement of the angle.

Angle $MJN=2x+10=2(5)+10=20^{\circ}$

The central angle is $2$ times the measure of the inscribed angle that intercepts the same arc.

20\times{2}=40^{\circ}

23.5^{\circ}

47^{\circ}

94^{\circ}

43^{\circ}

Angle $BDC$ is a central angle of circle $D$ because the vertex is at the center of the circle and the sides are radii.

Angle $BAC$ is an inscribed angle because the vertex is on the edge of the circle and the sides are chords.

Triangle $BCD$ has interior angles that sum to $180^{\circ}.$

So, to find the central angle, angle $BDC,$ subtract $59^{\circ}$ and $74^{\circ}$ from $180^{\circ}.$

180-(59+74)=47^{\circ}

The inscribed angle is $\cfrac{1}{2}$ the measure of the central angle that intercepts the same arc.

Angle $BAC=\cfrac{1}{2}\times{47}=23.5^{\circ}$

Central angle FAQs

What is the difference between a major arc and a minor arc?

A major arc is an arc that has a measure between $180^{\circ}$ and $360^{\circ}.$ A minor arc is an arc that has a measure between $0^{\circ}$ and $180^{\circ}.$

What is the arc length?

Arc length is part of the circumference of the circle. The circumference of the circle is the distance around the circle so the length of the arc is a portion of the circumference of a circle.

What are radians?

Radians are a unit of measure for an angle. Typically, in trigonometry, angle measures will be expressed in radians.

What are congruent arcs?

Congruent arcs are arcs that have the same length and the same measurement. If the inscribed angles of a circle are congruent and intercept arcs, those arcs will be congruent.

What are secants?

Secants are lines that intersect a circle at two points of intersection. Chords are line segments that are created by secant lines.

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