Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying by
Consider two circles and their respective diameters and circumferences.
In this diagram, is a central angle of the circle. Note that the measure of the intercepted arc of a central angle is equal to the measure of the central angle.
Recall that both a circle and a complete angle measure degrees. Since the vertex of a central angle lies at the center of a circle, both the central angle and its intercepted arc represent the same portion of the circle. Use the slider in the applet to see this relationship.
Therefore, the central angle and its intercepted arc have the same measure.
Suppose is drawn in circle such that it is a central angle. If is less than the endpoints of the angle, and create a minor arc on the circle. This arc can be named
If is greater than 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 In the diagram above,
The measure of an arc is equal to the measure of its corresponding central angle. In the diagram, the measure of is equal to the measure of
Suppose the measure of is given as The measure of the corresponding major arc can be found by subtracting from the measure of the entire circle.
Find the measure of arcs and
Find the length of arc
To begin, we can make sense of the given information. The circle has a radius of inches and the arc mesaures The relationship between arc measure and arc length can be used. First, the circumference of circle can be calculated as follows.The length of arc can be calculated by substituting the values for circumference and arc measure, and solving.
The measures of arcs and are equal because they are created by the same central angle. Because circle has a radius of the length of arc can be found using the definition of arc length. Because all circles are similar, circle circle Thus, the ratios of corresponding parts between the circles are proportional. An equation relating arc length to radius can be written. Simplifying the right-hand side and substituting the found value for the length of arc gives the following. The equation above shows that the length of arc is proportional to the radius of the circle. In fact, the expression 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 is the radian measure of a circle is radians — which corresponds to
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 radians. It is also known that this same distance can be expressed as Thus, Simplifying gives the following conversion factor.
Convert to radians and to degrees.
A circle with radius will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is the arc length of each semicircle is half this value,
Now, the above sectors will be unfolded. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. Therefore, the area of the figure below should be the same as the circle's area.
It was shown that the area of a circle is the product of and the square of its radius.
From the fact that equals an equivalent formula can be written if the central angle is given in radians.
Since the measure of an arc is equal to the measure of its central angle, the arc measures Therefore, by substituting for another version of the formula is obtained which can also be written in degrees or radians.
Consider sector bounded by and
Since a circle measures this sector represents of Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to
Recall that the area of a circle is By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained. Therefore, the area of a sector of a circle can be found by using the following formula.
Find the area of the sector.