arc

An arc is part of a circle's circumference.
 

Definition: In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc.

Definition: In a circle, the length of an arc is a portion of the circumference. Remembering that the arc measure is the measure of the central angle, a definition can be formed as:

Example: 

In circle O, the radius is 8, and the measure of minor arc is 110 degrees.  Find the length of minor arc to the nearest integer.Solution:    =  15.35889742 = 15

 

Understanding how an arc is measured makes the next theorems common sense.

Theorem: In the same circle, or congruent circles, congruent central angles have congruent arcs. Theorem: (converse) In the same circle, or congruent circles, congruent arcs have congruent central angles.

Remember:  In the same circle, or congruent circles, congruent arcs have congruent chords.  Knowing this theorem makes the next theorems seem straight forward.

Theorem:	In the same circle, or congruent circles, congruent central angles have congruent chords.
Theorem: (converse)	In the same circle, or congruent circles, congruent chords have congruent central angles.

The arc of a circle is a portion of the circumference of a circle. Measure an arc by two methods: 1) the measure of the central angle or 2) the length of the arc itself. Measure of the central angle: The XZ arc measures 120°. The XY arc measures 140°. The length of an arc (or arc length) is traditionally symbolized by s. The formula for finding arc length in radians is where r is the radius of the circle and θ is the measure of the central angle in radians. A comparison of degree and radian measure to find the arc length: Degrees Radians 90° π/2 60° π/3 45° π/4 30° π/6 From our previous course on radian , we realise the radian = .We will need this information to find the arc of a circle. Let us look first at how to find the arc of a circle by using its radian. Let us consider a circle with centre O and radius r. We compare between the lengths and angles subtended of the arc and circle. Hence the length of minor arc AB, Area of a Sector We shall look at how to calculuate the area of a sector ,using radian and radius. The area of a sector is also directly proportional to the angle within the sector. Example Before the start of doing or looking through any of these questions, it would be good to draw labelled diagrams to give you a better understanding. a) The radius of a circle with centre O is 5cm and the sector AOB subtends an angle of 2 radians. What is the length of its arc AB? = 5(2) = 10 cm The length of arc AB is 10 cm. b) The circle with centre O has radius 3 cm and the angle which sector COD subtends is . Find the area of sector COD. Remembering the formula for area of sector of a circle , A = = = 18cm Arc of a Circle An arc is any connected part of the circumference of a circle. In the diagram above, the part of the circle from B to C forms an arc. It is called arc BC. An arc could be a minor arc, a semicircle or a major arc. A semicircle is an arc that is half a circle. A minor arc is an arc that is smaller than a semicircle. A major arc is an arc that is larger than a semicircle. Central Angle A central angle is an angle whose vertex is at the center of a circle. In the diagram above, the central angle for arc BC is 45°. The sum of the central angles in any circle is 360°. Arc Measure The measure of a semicircle is 180°. The measure of a minor arc is equal to the measure of the central angle that intercepts the arc. We can also say that the measure of a minor arc is equal to the measure of the central angle that is subtended by the arc. In the diagram below, the measure of arc BC is 45°, The measure of the major arc is equal to 360° minus the measure of the associated minor arc. The following video shows how to identify semicircle, minor arc and major arc and their measures. <!-- google_ad_section_end --> <!-- google_ad_section_start(name=m2) --> Arc Length Formula The arc length is the distance along the part of the circumference that makes up the arc. Arc Measure given in Degrees Since the arc length is a fraction of the circumference of the circle, we can calculated it in the following way. Find the circumference of the circle and then multiply by the measure of the arc divided by 360°. Remember that the measure of the arc is equal to the measure of the central angle. The formula for the arc length of a circle is where r is the radius of the circle and m is the measure of the arc (or central angle) in degrees. Worksheet to calculate arc length and area of a sector (degrees). Arc Measure given in Radians If the measure of the arc (or central angle) is given in radians, then the formula for the arc length of a circle is the product of the radius and the arc measure. Arc Length = r × m where r is the radius of the circle and m is the measure of the arc (or central angle) in radians The above formulas allow us to calculate any one of the values given the other two values. Worksheet to calculate arc length and area of sector (radians). Calculate Arc Length given Measure of Arc in degrees From the formula, we can calculate the length of the arc. Example: If the circumference of the following circle is 54 cm, what is the length of the arc ABC? Solution: Circumference = 2πr =54 × = 18 cm Example: If the radius of a circle is 5 cm and the measure of the arc is 110˚, what is the length of the arc? Solution: This video shows how to define arc length and how it is different from arc measure. It will also show how to calculate the length of an arc when the arc measure is given in degrees. This video shows how to use the Arc Length Formula when the arc measure is given in degrees. The following video shows how to find the arc length on a circle when the central angle is given in degrees. Calculate Arc Length given Measure of Arc in radians If the measure of the arc (or central angle) is given in radians, then the formula for the arc length of a circle is Arc Length = r × m where r is the radius of the circle and m is the measure of the arc (or central angle) in radians This video shows how to use the Arc Length Formula when the measure of the arc is given in radians. Definition of Arc Length and Finding the Arc Length when the central angle is given in radians.