Central Angle on Sector

How a Central Angle Sets the Size of a Sector

A circle sector is bounded by two radii and the arc between them. The two radii form a central angle whose vertex lies at the center of the circle. An open hand fan or a slice of cake has the same basic shape.

The central angle determines what fraction of the circle the sector occupies. For a fixed radius, a larger angle produces a larger sector area, while a smaller angle produces a smaller one.

Central Angle Forming a Sector

The central angle is shown with vertex at center

O

and two radii as the angle sides.

Properties of Central Angle in Sectors

Both angle sides: Always consist of radii of equal length
Angle vertex: Always located exactly at the center of the circle
Proportion to circle: The size of the central angle determines the sector's proportion to the entire circle

Within the same circle, equal central angles produce equal sector areas. Dividing the full angle into equal angles therefore divides the circle into equal areas.

Relationship Between Central Angle and Sector Area

A full turn corresponds to the whole circle area. For a sector, the area comes from the central angle, and the angle unit decides which formula applies:

For angles in degrees:

A = \frac{\theta}{360^\circ} \times \pi r^2

For angles in radians:

A = \frac{1}{2} \theta r^2

The variables in both formulas are:

$A$ = sector area
$\theta$ = central angle, measured in degrees in the first formula and radians in the second
$r$ = radius

Both formulas show that sector area is proportional to the central angle size. When the central angle is doubled, the sector area also becomes twice as large. Conversely, if the central angle is halved, the sector area becomes half.

Comparison of Central Angles on Different Sectors

Visualization of sectors with central angles of

45^\circ

90^\circ

, and

135^\circ

showing proportional relationships.

Measurement and Identification of Central Angles

To identify a central angle, check that its vertex lies at the center of the circle and that both sides are radii. By contrast, an inscribed angle has its vertex on the circumference.

Central angle measurement methods:

Using a protractor: Place the protractor's center point exactly at the circle's center, ensure the base line is parallel to one radius, then read the scale at the position of the second radius
Mathematical calculation: Using relationships among central angle, radius, arc length, and sector area
Circle division: dividing $360^\circ$ by the number of equal parts

Once the central angle is known, its value can be used to calculate the corresponding sector area or arc length.

Central Angle Measurement Technique

Demonstration of measuring a

60^\circ

central angle with clear bounding radii.

Central Angle and Arc Theorem

For a central angle and its intercepted arc, the following relationship holds:

Main theorem:

The central angle measure equals the arc measure it subtends
If a central angle measures $\alpha$ degrees, then the arc it forms also measures $\alpha$ degrees

Properties:

In the same circle, congruent central angles produce congruent arcs and sectors
If two sectors of the same circle have the same area, then their central angles have the same measure
When adjacent central angles are added, the measures of their corresponding arcs are added as well

These properties connect central angles, arc measures, and sector areas within one circle. Always distinguish an arc's degree measure from its physical length.

Calculating Central Angle from Sector Information

Determining the central angle measure can be done through various known information about the sector:

From sector area and radius:

$\alpha = \frac{360^\circ \times A}{\pi r^2}$
From arc length and radius:

$\alpha = \frac{180^\circ \times s}{\pi r}$
From equal circle division:

$\alpha = \frac{360^\circ}{n}$

where:

$\alpha$ = central angle (in degrees)
$A$ = sector area
$s$ = arc length
$r$ = radius
$n$ = number of divisions

Choose the formula that matches the given information: sector area, arc length, or the number of equal parts.

Central Angle Parameters

Visualization of elements: radius

r

, arc length

s

, and central angle

72^\circ

Formula application examples:

Round pizza cut radially into $8$ equal sectors: $\alpha = \frac{360^\circ}{8} = 45^\circ$
Adjacent hour marks on a clock face with $12$ equal divisions: $\alpha = \frac{360^\circ}{12} = 30^\circ$
Round cake cut radially into $6 \text{ equal sectors}$ : $\alpha = \frac{360^\circ}{6} = 60^\circ$

These examples demonstrate the direct application of circle division formulas in daily life, where each part has an equal central angle.