An angle whose vertex lies at the center of a circle is called a central angle.
Suppose ∠ACB is drawn in circle C such that it is a central angle. If m∠ACB is less than 180∘, the endpoints of the angle, A and B, create a minor arc on the circle. This arc can be named $\Arc{AB}.$
If m∠ACB is greater than 180∘, the endpoints of the angle create a major arc on the circle. In other words, the points on the circle not included in the minor arc create a major arc.
A major arc can be named using its endpoints and another point on the arc. Therefore, the major arc can be named $\Arc{ADB}.$ In the diagram above,
The measure of an arc is equal to the measure of the central angle that creates it. Suppose arc AB is created when the central angle ∠ACB is drawn inside circle C. In the diagram above, the measure of arc AB=m∠C.
Suppose the measure of arc AB is given as θ. The measure of the corresponding major arc, arc ADB, can be found by subtracting θ from 360∘, the measure of the entire circle.
Arc length measures the portion of the circle's circumference an arc spans. In the diagram below, the measure of arc AB equals θ, while the length of arc AB corresponds to the portion of the circumference the red arc spans.
The ratio of an arc's length to the circumference of the circle is equal to the ratio of the arc's measure to 360∘.
circumferencearc length=360∘arc measure
Substituting θ for the arc measure and 2πr for the circumference gives the following.
2πrarc length=360∘θFind the measure of arcs AB and BCA.
Find the length of arc AB.
To begin, we can make sense of the given information. The circle C has a radius of 4 inches and the arc AB mesaures 50∘. The relationship between arc measure and arc length can be used. circumferencearc length=360∘arc measure First, the circumference of circle C can be calculated as follows. CCC=2πr=2π⋅4=8π
The length of arc AB can be calculated by substituting the values for circumference and arc measure, and solving.In the diagram below, circle AB has a radius of 1 unit, and circle CD has a radius of r units.
The measures of arcs AB and CD are equal because they are created by the same central angle. Because circle AB has a radius of 1, the length of arc AB can be found using the definition of arc length. arc length AB=360∘arc measure AB⋅2π Because all circles are similar, circle AB∼ circle CD. Thus, the ratios of corresponding parts between the circles are proportional. An equation relating arc length to radius can be written. arc length ABarc length CD=1r Simplifying the right-hand side and substituting the found value for the length of arc AB gives the following. arc length CD=r⋅arc length AB=r⋅(360∘arc measure AB⋅2π) The equation above shows that the length of arc CD is proportional to the radius of the circle. In fact, the expression 360∘arc measure AB⋅2π
is called the constant of proportionality, and defines the radian measure of the central angle associated with the arc. Because the circumference of a circle with radius 1 is 2π, the radian measure of a circle is 2π radians — which corresponds to 360∘.Because angles can be measured in both radians and degrees, it can be useful to translate between the two. The concept of radian measure asserts that one revolution around a circle corresponds to exactly 2π radians. It is also known that this same distance can be expressed as 360∘. Thus, 2π radians=360∘. Simplifying gives the following conversion factor.
π radians=180∘
Convert 45∘ to radians and 2π radians to degrees.
A sector of a circle is a portion of the circle enclosed by two radii and an arc.
The area of a sector of a circle can be determined using the proportional relationship between the circle's and the sector's areas and measures. area of circlearea of sector=measure of circlemeasure of arc AB Substituting the known values gives the following. πr2area of sector=360∘θ Therefore, the area of a sector of a circle can be found using the following formula.
area of sector=360∘θ⋅πr2
Find the area of the sector.