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## What is a Central Angle? – Definition, Theorem & Formula

A central angle is an angle comprised of two radii that meet at the center of a circle. Learn the definition and theorem of a central angle, and how to use the formula to find a central angle when provided the radius and arc length of a circle.

## Definition Of A Central Angle

A **central angle** is the angle that forms when two radii meet at the center of a circle. Remember that a vertex is the point where two lines meet to form an angle. A central angle’s vertex will always be the center point of a circle.

The angle subtended by an arc at the center of a circle is the central angle. The radius vectors form the arms of the angle. In other words, an angle whose vertex is the center of a circle with the two radii lines as its arms that intersect at two different points. When these two points are joined they form an arc. Central angle is the angle subtended by this arc at the center of the circle.

Here O is the center of the circle, AB is the arc and, OA is a radius and OB is another radius of the circle.

Here “s” is the length of the arc and “r” is the radius of the circle. This is the formula for finding central angle in degrees. For finding the central angle in radians, we have to divide the arc length by the length of the radius of the circle.

## Reflex Versus Convex

It’s important to know that when two radii meet at the center of a circle to create a central angle, they also create another angle in the process. The **convex central angle** is the one that is shown in this diagram.

A convex central angle is a central angle that measures less than 180°.

The **reflex central angle** that is formed is on the other side of the convex angle. Reflex angles measure more than 180°

and less than 360°. Check out this second image to notice the reflex angle that is created on the other side of the convex central angle.

## Central Angle Theorem

Before we understand what the central angle theorem is, we must understand what subtended and inscribed angles are, because they are a part of the definition. A **subtended angle** is an angle that is created by an object at a given outer position.

If you can’t wrap your head around that definition, picture this: You are standing on planet Earth and looking up at the sun. A triangle with three sides is formed. The three sides are the ray of the sun that goes from the top of the sun to your eyes, the ray that goes from the bottom of the sun to your eyes, and the height of the sun.

We would say that the angle subtended by the sun to your eyes changes, although not by much, depending on if you move forward or backward. The angle will increase in size if you step forward and decrease if you step backward.

An **inscribed angle** is one that is subtended at a point on a circle by two identified points on a circle.

If you can’t wrap your head around that definition, picture this: Imagine that you are jogging around a perfect circular pond in your neighborhood. There are two neighbors, Ed and Tom, on the opposite side of the pond, close to each other, but at different points. Now picture a line drawn from Ed to you and one from Tom to you. The angle that is created when both lines are drawn to you is an inscribed angle, because all three points are directly on the circle.

angle, because all three points are directly on the circle.

*The angle at your location is the inscribed angle.*

Now that you understand what subtended and inscribed angles means, we can move forward to the Central Angle Theorem.

The **Central Angle Theorem** states that the central angle subtended by two points on a circle is always going to be twice the inscribed angle subtended by those points.

In the three diagrams you can see now, you can see the inscribed angle ADB is always half the measurement of the central angle ACB, no matter where the vertex of the angle (point D) is on the circle.

The Central Angle Theorem is always true unless the vertex of the inscribed angle (point D) lies on the minor arc instead of the major arc.

Let’s look at the major arc vs. minor arc first:

The length between points A and B is called the arc length (we will need this information later).

Now that we know the difference between the major and minor arcs, we can use the following formula to determine the measurement of the inscribed angle when the vertex of the inscribed angle (point D) lies on the minor arc instead of the major arc.

## What is a central angle?

A **central angle** is an angle with a vertex at the centre of a circle, whose arms extend to the circumference. You can imagine the central angle being at the tip of a pizza slice in a large circular pizza.

You can find the central angle of a circle using the formula:

`θ = L / r`

where **θ** is the central angle in radians, **L** is the arc length and **r** is the radius.

## Where does the central angle formula come from?

The simplicity of the central angle formula originates from the definition of a radian. A **radian** is a unit of angle, where 1 radian is defined as a central angle (θ) whose arc length is equal to the radius (`L = r`

).

## The circle angle calculator in terms of pizza

Because maths can make people hungry, we might better understand the central angle in terms of pizza. What would the central angle be for a slice of pizza if the crust length (L) was equal to the radius (r)?

Since the problem defines `L = r`

, and we know that 1 radian is defined as the central angle when L = r, we can see that the central angle is 1 radian. We could also use the central angle formula as follows:

`θ = L / r`

`= L / L`

`= 1 rad`

## How many pizza slices with a central angle of 1 radian could you cut from a circular pizza?

In a complete circular pizza, we know that the central angles of all the slices will add up to 2π radians = 360°. Since each slice has a central angle of 1 radian, we will need `2π / 1 = 2π`

slices, or 6.28 slices to fill up a complete circle.

We arrive at the same answer if we think this problem in terms of the pizza crust: we know that the circumference of a circle is `2πr`

. Since the crust length = radius, then `2πr / r = 2π`

crusts will fit along the pizza perimeter.

Now, if you are still hungry, take a look at the sector area calculator to calculate the area of each pizza slice!

## Bonus challenge – How far does the Earth travel in each season?

Try using the central angle calculator in reverse to help solve this problem. The Earth is approximately 149.6 million km away from the Sun. If the Earth travels about one quarter of its orbit each season, how many km does the Earth travel each season (e.g., from spring to summer)?

Let’s approach this problem step-by-step:

- Simplify the problem by assuming the Earth’s orbit is circular (
*The Earth’s orbit is actually elliptical, and constantly changing*). In this model, the Sun is at the centre of the circle, and the Earth’s orbit is the circumference. - The
**radius**is the distance from the Earth and the Sun: 149.6 million km. - The
**central angle**is a quarter of a circle:`360° / 4 = 90°`

. - Use the central angle calculator to find
**arc length**.

You can try the final calculation yourself by rearranging the formula as:

`L = θ * r`

Then convert the central angle into radians: `90° = 1.57 rad`

, and solve the equation:

`L = 1.57 * 149.6 million km`

`L = 234.9 million km`

.

**When we assume that for a perfectly circular orbit, the Earth travels approximately 234.9 million km each season!**

# Central Angle of a Circle Formula

The angle between two radii of a circle is known as the central angle of the circle. The two points of the circle, where the radii intersect in the circle (Note – The other end of the radii meets at the centre of the circle), forms a segment of the Circle called the Arc Length.

## Central Angle

Central Angle is the angle formed by two arms with the center of a circle as the vertex. The two arms form two radii of the circle intersecting the arc of the circle at different points. Central angle helps to divide a circle into sectors. A slice of pizza is a good example of central angle. A pie chart is made up of a number of sectors and helps to represent different quantities. A protractor is a simple example of a sector with a central angle of 180º. Central angle can also be defined as the angle formed by an arc of the circle at the center of the circle.

## How To Find Central Angle?

The central angle is the angle between any two radii of a circle. To find the central angle we need to find the arc length (which is the distance between the two points of intersection of the the two radii) and the radius length. The steps given below shows how to calculate central angle in radians.

There are three simple steps to find the central angle.

- Identify the ends of the arc and the center of the circle (curve). AB is the arc of the circle and O is the center of the circle.

Join the ends of the arc with the center of the circle. Also, measure the length of the arc and the radius. Here AB is the length of the arc and OA and OB are the radii of the circle.

Divide the length of the curve with the radius, to get the central angle. By using the formula shown below, we will find the value of the central angle in radians.

**Important Notes**

- The central angle of a circle is measured in radian measure and sexagesimal measure.
- The unit of radian measure is radians and the unit of sexagesimal measure is degrees.
- Radian × (180/π) = Sexagesimal

## Formulas

## Solved Examples

**Example 1: Sam measures the angle in a triangle with the help of a protractor as 60º. Convert the angle into radian measure.**

Solution:

The given angle of 60° is in sexagesimal measure.

Radian = π/180° × Sexagesimal

Radian = π/180° × 60°

Radian = π/3

Therefore, the angle is π/3 radians.

**Example 2: Larry drew a circle and cut it into four equal parts using two diameters. How can you help Larry to measure the central angle or inscribed angle of each part of the circle?**

Solution:

Larry cuts the circle into four equal parts.

Complete angle in a circle = 360°

Angle of each quadrant = 360°/4

= 90°

Therefore, the central angle of a quadrant is 90°.

**Example 3: Sally marks an arc of length 8 inches and measures its central angle as 120 degrees. What is the radius of the arc?Solution: **

Radius of the arc = 8 inches

Central Angle = 120°

Central angle = (length of arc × 360°)/(2 π × radius)

radius = (length of arc × 360°)/(2 π × Central angle)

radius =(8 × 360°) / (2 π × 120°)

radius = 12/π

Therefore, the radius is 12/π inches.

**Example 4: Jim uses a compass to draw an arc of length 11 inches and a radius of 7 inches. Without using a protractor, how can Jim calculate the angle of this arc?**

Solution:

Length of the arc = 11 inches

Radius of the arc = 7 inches

Angle of the arc = (length of arc × 360°)/(2 π r)

Angle = (11 × 360°)/ (2 × 22/7 × 7)

Angle = 90°

Therefore, the angle of the arc is 90°.

**Example 5: George wants to create a garden in the shape of a sector of radius 42 feet and having a central angle of 120 degrees. Calculate the area of the grass which is required to cover the garden. ****Solution: **

Given that the shape of the garden is a sector.

Radius = 42 feet

Central Angle = 120°

The area of the grass required to cover the garden is the same as the area of the sector.

Area of the sector = θ/360° × π r^{2}

Area =(120°/360°) × π × 42^{2}

Area = 1/3 × 22/7 × 42 × 42

Area = 22 × 2 × 42

Area = 1848

Therefore, the area of the sector is 1848 square feet.

## FAQs on Central Angle

### How Do You Find the Central Angle of an Arc?

To find the central angle of an arc, connect the ends of the arc with the center of the circle using the radius vectors. The angle between the two radii represents the central angle of the arc.

### What is the Difference Between Reflex and Convex Angles?

Both reflex angle and convex angle can be central angles of a circle. A reflex angle is greater than 180 degrees and less than 360 degrees. A convex angle is less than 180 degrees.For a given arc of a circle, the sum of its convex angle and reflex angle is equal to the complete angle. A complete angle is equal to 360 degrees. Convex angle + Reflex angle = Complete angle

### What Is Central Angle Definition in Geometry?

As per central angle definition in geometry, it is the angle subtended by the arc of the circle at the center of the circle. The two radii make the arms of the angle.

### How Many Degrees is the Central Angle of a Circle?

The degrees of a central angle is the angle made by the arc at the center of the circle.

### What is The Central Angle of a Curve?

The central angle of a curve is the angle subtended by it at the center of the curve.

### What is Central Angle Theorem?

The central angle theorem states that the angle subtended by an arc at the center of the circle is double the angle subtended at any point on the circumference of the circle.

### How Do You Measure Central Angle of a Circle?

The central angle of a circle is measured in either degrees or radians. It is measured with the help of length of the arc and length of the radius of the circle. The formula to measure central angle (in radians) = (Length of the arc)/(Length of the radius).

### What is the Central Angle Made by a Semi-circle?

The central angle made by a semi-circle is 180°.

### What is an Inscribed Angle?

The angle subtended by an arc at any point on the circle is called an inscribed angle.

### What is the Difference Between Central Angle and Inscribed Angle?

Central angle is the angle subtended by an arc at the center of a circle. Inscribed angle is an angle subtended by an arc at any point on the circumference of a circle.

### What is the Central Angle Equal To?

Answer) The arc length divided by the radius of the circle is the measure of an arc. In radians, the arc measure equals the corresponding central angle measure. That is why radians are so natural: a central angle of one radian spans an arc that is exactly one radius long.

**Is 180 a Major Arc?**

Answer) A minor arc is less than 180 degrees in length. A major arc is defined as an arc with a length greater than 180 degrees. Because it divides the circle in half, an arc with a measure of 180 degrees is known as a semicircle. The vertex of a central angle is generally the centre of a circle.

**MATHS Related Links**

Central Angle of a Circle Formula

Surface Area of Circle Formula

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