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## Central Angle Definition

A central angle is a vital concept in geometry, especially when studying circles. It’s an angle whose vertex is located at the centre of a circle and whose sides (rays) extend to the circumference. Grasping the definition and properties of a central angle can help you understand more complex geometrical notions.

### Understanding the Central Angle

A central angle is formed by two radii of a circle. The measure of a central angle can offer insights into the arc that it subtends, meaning the portion of the circle's circumference that lies between the two points where the angle's sides meet the circle.

**Central Angle**: An angle whose vertex is at the centre of a circle and whose arms (sides) are radii that extend to the circle’s circumference.

If you have a circle with centre **O** and points **A** and **B** on the circumference, then the angle **∠AOB** is a central angle.

A full circle is 360 degrees, so the sum of all central angles in a circle is also 360 degrees.

### Measuring a Central Angle

The measure of a central angle is directly equal to the measure of the arc it intercepts. This means a central angle measuring 60 degrees subtends an arc that measures 60 degrees.

- A central angle measuring 90 degrees intercepts a quarter (or a 90-degree arc) of the circle.
- A central angle of 180 degrees intercepts a semicircle (or a 180-degree arc).

Consider a circle divided into equal sectors. If there are 6 sectors, then each central angle measures:

\[\frac{360°}{6} = 60°\]

Each sector thus corresponds to a 60-degree arc of the circle.

**Fun Fact:** Central angles also have a unique relationship with the concept of radians. One radian is the measure of a central angle that intercepts an arc equal in length to the circle’s radius. This connects the concepts of degrees and radians in angular measurement.

## Central Angle of a Circle

A central angle is fundamental when studying circles. It’s an angle that has its vertex at the centre of the circle with its sides (rays) touching the circumference. Understanding the properties of a central angle will enhance your comprehension of more complex geometrical concepts.

### Properties of a Central Angle

A central angle is formed by two radii of a circle. The measure of this angle informs about the arc it subtends, meaning the part of the circle's circumference that lies between the two points where the angle's rays intersect the circle.

- A central angle measuring
**90 degrees**subtends a quarter circle arc. - A central angle of
**180 degrees**subtends a semicircle.

**Central Angle:** An angle whose vertex is at the centre of a circle and whose arms (sides) are radii that extend to the circle’s circumference.

If a circle has centre **O** and points **A** and **B** on the circumference, then the angle **∠AOB** is a central angle. For example, if **∠AOB** is 60 degrees, it means the arc **AB** also measures 60 degrees.

Consider if there are **7** sectors in a circle, the measure of each central angle can be calculated: \[\frac{360°}{7} ≈ 51.43°\]

Remember, a full circle measures 360 degrees, which means the sum of all central angles around a single point is also 360 degrees.

### Measuring a Central Angle

The measure of a central angle is equal to the length of the arc it intercepts when expressed in degrees. This relationship is straightforward because a central angle and its intercepted arc share the same angular measure.

- If a central angle is
**30 degrees**, the intercepted arc is also**30 degrees**. - A central angle of
**120 degrees**will subtend an arc of**120 degrees**.

Consider a circle partitioned into equal 8 sectors. To find the central angle of each sector: \[\frac{360°}{8} = 45°\]

**Fun Fact:** Central angles are also linked to the concept of radians. A radian is the measure of a central angle that intercepts an arc with a length equal to the circle’s radius. One full circle measures \[2π\] radians, which equals 360 degrees.

## How to Find Central Angle

Determining the measure of a central angle requires an understanding of the circle's properties and the relationship between the central angle and its corresponding arc. The process often involves simple division and can be made easier by applying specific formulas.

### Central Angle Formula

The formula for calculating a central angle depends on the proportion of the circle's circumference corresponding to the given arc. If you know the length of the arc and the radius of the circle, you can use the central angle formula:

**Central Angle (in degrees)**= \( \frac{{\text{Arc Length}}}{{\text{Circumference}}} \times 360 \)**Arc Length**= Circumference \( \times \frac{{\text{Central Angle (in degrees)}}}{360} \)

Circumference of a circle is calculated as \(2 \pi r\) where \(r\) is the radius of the circle.

For a circle with radius 5 cm and an arc length of 10 cm, calculate the central angle:

Circumference = \(2 \pi \times 5 = 10 \pi\)**Central Angle** = \( \frac{10}{10\pi} \times 360 \approx 114.6^{\circ} \)

**Historical Insight:** The concept of radians, widely used in trigonometry, originates from central angles. One radian is defined as the angle subtended by an arc equal in length to the radius of the circle. This means that in a circle, there are \(2\pi\) radians, equivalent to 360 degrees. Hence, one radian equals \( \frac{180^{\circ}}{\pi} \) or approximately 57.3 degrees.

## Central Angle Theorem

The Central Angle Theorem is a significant geometrical principle that establishes a relationship between the central angle and the arc it subtends. This theorem is vital in various geometrical problems, including those involving circles and their properties.

### Statement of the Theorem

The Central Angle Theorem states that the measure of a central angle is equal to the measure of the arc it intercepts. This theorem can be visualised easily by considering how the angle and its arc share the same origin at the circle's centre.

**Central Angle Theorem:** The theorem states that the measure of a central angle is exactly equal to the measure of the arc that it intercepts.

## Central Angle - Key takeaways

**Central Angle Definition**: An angle whose vertex is at the centre of a circle and whose arms (sides) are radii that extend to the circle’s circumference.**Measuring Central Angles**: The measure of a central angle is directly equal to the measure of the arc it intercepts.**Central Angle Theorem**: States that the measure of a central angle is exactly equal to the measure of the arc that it intercepts.**Central Angle Formula**: Central Angle (in degrees) = (Arc Length / Circumference) x 360.**How to Find Central Angle**: Involves using the formula based on the arc length and the circle's radius or circumference.

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