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# Central Angles

A central angle is an angle with its vertex located at the center of a circle and its sides containing two radii of the circle. The two arms of the angle intersect the arc of the circle at different points, forming two radii of the circle:

In the diagram, there are three central angles: $\angle \mathrm{PZQ}$ , $\angle \mathrm{QZR}$ , and $\angle \mathrm{RZP}$ . Central angles are useful in dividing a circle into sectors. A slice of pizza is a good example of a central angle in action. Additionally, a pie chart is made up of multiple sectors, each represented by a central angle, and is used to represent different quantities.

## The sum of central angles

A central angle can be defined as the angle formed by an arc of the circle at the center of the circle. The sum of the measures of the central angles of a circle with no points in common is 360°.

For example, in the figure above, $m\angle \mathrm{PZQ}+m\angle \mathrm{QZR}+m\angle \mathrm{RZP}=360°$ .

Example 1

Find the value of x.

To find the value of x, consider that the sum of the measures of the central angles in a circle with no interior points in common is equal to 360°. Therefore, we can write the equation:

$m\angle \mathrm{MON}+m\angle \mathrm{NOP}+m\angle \mathrm{POM}=360°$

Substituting known values, we have:

$50°+90°+x=360°$

Simplifying, we get:

$140°+x=360°$

Subtracting 140° from each side, we find that

$x=220°$

Example 2

Sheryl drew a circle and divided it into four equal parts using two diameters. To measure the central angle of each part of the circle, divide 360° by 4. The result is 90°, which represents the central angle of each section.

Example 3

Luiz has a pie chart indicating how much time he spends on each subject every night for homework. The total of the sections equals 360°. The central angle of the following subjects is as follows:

• Math: 38°
• Science: 44°
• Social Studies: 18°
• English: 80°
• French: 90°

He also spends time each night practicing his instrument, which takes up the rest of the pie chart. What is the central angle of the section of the pie chart that takes up his time practicing his instrument?

We know that the entire pie chart is 360°

$38°+44°+18°+80°+90°+x=360°$

Simplify the problem by doing the math you can do so far.

$270°+x=360°$

Subtract 270° from each side.

$x=90°$

The section of the pie chart that indicates how much time Luiz spends practicing his instrument is 90°, the same amount of time he spends working on French homework.

Arcs