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Arc Measures

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It is very important to be familiar with the anatomy of a circle and especially the angles within it. This article covers the properties of **arc measures**, the formula for an arc measure, and how to find it within a geometric context.

There are two important definitions to be aware of:

An **arc** is the edge of a circle **sector**, i.e. the edge bounded/delimited by two points in the circle.

**Arc length **is the size of the arc, i.e. the distance between the two delimiting points on the circle.

If we think of an **arc** as being the edge between two points A and B on a circle, the **arc measure **is the size of the angle between A, the centre of the circle, and B.

In relation to the arc length, the arc measure is the size of the angle from which the arc length subtends.

Here are these definitions demonstrated graphically:

Before we introduce the formula for arc measurement, let’s recap **degrees** and **radians**.

**To convert radians to degrees**: divide by and multiply by 180.

**To convert degrees to radians**: divide by 180 and multiply by.

Here are some of the common angles which you should recognise.

Degrees | 0 | 30 | 45 | 60 | 90 | 120 | 180 | 270 | 360 |

Radians | 0 |

The formula that links both the arc measure (or angle measure) and the arc length is as follows:

Where

*r*is the radius of the circle- is the arc measure in radians
*S*is the arc length

We can find the arc measure given the radius and the arc length by rearranging the formula: .

Find the arc measure shown in the following circle in terms of its radius, *r*.

Using the formula :

We need the arc measure in terms of *r*, so we need to rearrange this equation~~:~~

If we are not given the radius, *r*, then there is a second method for finding the arc measure. If we know the circumference of a circle as well as the arc length, then the **ratio** between the **arc measure** and (or depending on whether you want the arc measure in degrees or radians) is equal to the ratio between the **arc length**** **and the** circumference.**

Where

*c*is the circumference of the circle- is the arc measure in
**degrees** *S*is the arc length

Find the arc length, x, of the following circle with a circumference of 10 cm.

Using the formula :

Rearranging, we get:

to 3 s.f.

- An
**arc**is the edge of a circle**sector**, i.e. the edge bounded/delimited by two points in the circle. **Arc length**is the size of the arc, i.e. the distance between the two delimiting points on the circle.- An arc measure is the size of the angle from which the arc subtends.
- Finding the arc measure given the radius and arc length:
Where

*r*is the radius of the circle.- is the arc measure in radians.
*S*is the arc length.

Finding the arc measure given the circumference and arc length:

Where:

*c*is the circumference of the circle.- is the arc measure in degrees.
*S*is the arc length.

An arc measure is the angle from which an arc of a circle subtends.

The arc measure is the arc length divided by the radius.

The arc measure is the arc length divided by the radius.

In geometry, the arc measure is the arc length divided by the radius.

More about Arc Measures

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