Observe Figure 1. Given a point {eq}A {/eq}, a circumference is a set of all points that are at a fixed distance {eq}r {/eq} to {eq}A {/eq}, with the corresponding circle being the set of all points whose distance to {eq}A {/eq} is less than {eq}r {/eq}. Sometimes, though, the terms "circumference" and "circle" are used interchangeably.
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Figure 1 displays a circumference centered at {eq}A {/eq}. The distance from the center to any point on the circumference is called the radius. Any segment whose endpoints are the center of the circle and a point on the circumference is also called a radius of the circle. In other words, the radius may refer to both a length and a segment. In Figure 1, {eq}AB, AC {/eq} and {eq}AD {/eq} (or their length) are radii of the circumference.
A chord is any segment whose endpoints are on the circumference. Thus, {eq}EB {/eq} and {eq}CB {/eq} are two chords depicted in Figure 1. Note that chord {eq}CB {/eq} contains the center {eq}A {/eq}.
In this case, {eq}CB {/eq} is said to be the diameter of the circle. Like the radius, diameter also refers interchangeably to the length of a segment or to the segment itself. A diameter will always be twice the length of a radius and it is the biggest chord in a circle.
Other relevant terms in this context are arc, segment of a circle, and circular sector. An arc is a portion of the circumference. Segment {eq}EB {/eq}, for example, determines two arcs of the circumference: one that contains points {eq}C {/eq} and {eq}D {/eq}, and one that does not contain such points. Arcs are usually denoted by {eq}\overset{\frown}{EB} {/eq} or {eq}\overset{\frown}{ECB} {/eq}. The second example is the arc whose endpoints are {eq}E {/eq} and {eq}B {/eq} and that contains {eq}C {/eq}. This way it is possible to distinguish the two arcs determined by {eq}E {/eq} and {eq}B {/eq}.
The segment of a circle definition and the concept of circular sector involve arcs of circumference and segments of the line. A circular sector is the region of a circle delimited by two radii and the arc of circumference subtended by them. In Figure 1, for instance, the region that resembles a slice of pizza and is delimited by radii {eq}AC {/eq} and {eq}AD {/eq} and the arc {eq}\overset{\frown}{CD} {/eq} (the portion that does not contain {eq}E {/eq} or {eq}B {/eq}) is a circular sector. The segment of a circle, the main topic of this lesson, will be addressed in the following section.
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Try it nowWhat is a segment of a circle? It is the region of a circle bounded by a chord and an arc subtended by it. In Figure 1, chord {eq}EB {/eq} divides the circle into two parts: one that contains the center and one that does not. The bigger segment of a circle (in terms of area) is the major segment, whereas the other part is the minor segment. If the chord in question is the diameter, then both segments of a circle will have the same area and will be semi-circles.
In this section, it will be explained how to find the area of a segment of a circle. To this end, it will be necessary to recall the area of a circular sector and one particular triangle area formula.
Consider a triangle with two of the sides measuring {eq}a {/eq} and {eq}b {/eq}, and let the angle formed by those sides be {eq}\alpha {/eq}. The area of the triangle is given by {eq}A_t~=~\frac{1}{2}ab\sin\alpha {/eq}.
The area of a circular sector, in turn, is proportional to the central angle, that is, the angle formed by the radii that are part of the sector. Consider a circular sector where the radius is {eq}r {/eq} and the central angle is {eq}\theta {/eq}, measured in degrees. Using the relation $$\frac{\pi r^2}{360}~=~\frac{A_s}{\theta} $$ gives that the area of the circular sector is {eq}A_s~=~\frac{\pi r^2\theta}{360} {/eq}. If {eq}\theta {/eq} is measured in radians, replacing the {eq}360 {/eq} by {eq}2\pi {/eq} gives {eq}A_s~=~\frac{r^2\theta}{2} {/eq}.
To evaluate the area of a segment of a circle, analyze a few cases. In all of them, assume that the radius is {eq}r {/eq} and the central angle is {eq}x {/eq}. Figure 2 offers a visual aid.
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First, if the segment is a semi-circle, then its area will be {eq}A_{seg}~=~\frac{\pi r^2}{2} {/eq}. Otherwise, there will be a major or a minor segment. The minor segment is a circular sector minus a triangle. The area of the minor segment of a circle formula is $$A_{min~seg}~=~\frac{\pi r^2x}{360}~-~\frac{r^2\sin x}{2} $$ Equivalently, if the central angle is given in radians, the segment of a circle formula to determine the area of the minor segment is $$A_{min~seg}~=~\frac{r^2x}{2}~-~\frac{r^2\sin x}{2} $$
The corresponding major segment has area $$A_{maj~seg}~=~\pi r^2~-~\frac{\pi r^2x}{360}~+~\frac{r^2\sin x}{2} $$ for {eq}x {/eq} given in degrees and $$A_{maj~seg}~=~\pi r^2~-~\frac{r^2x}{2}~+~\frac{r^2\sin x}{2} $$ for {eq}x {/eq} given in radians.
Evaluate the area of a minor segment of a circle whose radius is {eq}3 {/eq} cm and corresponding central angle is {eq}\frac{\pi}{3} {/eq}.
Plugging the values in the formula gives {eq}A~=~\frac{\pi3^2\pi}{(3)2\pi}~-~\frac{3^2\sin\pi/3}{2}~=~\frac{3\pi}{2}~-~\frac{9\sqrt{3}}{4} {/eq} cm{eq}^2 {/eq}.
Consider the same circle as in Example 1, but find the major segment of the circle area using that the central angle is {eq}60^{\circ} {/eq}.
Using the formula gives {eq}A~=~\pi3^2~-~\frac{\pi3^260}{360}~+~\frac{3^2\sin60}{2}~=~9\pi~-~\frac{3\pi}{2}~+~\frac{9\sqrt{3}}{4}~=~\frac{15\pi}{2}~+~\frac{9\sqrt{3}}{4} {/eq} cm{eq}^2 {/eq}.
In this section, two theorems involving the segment of a circle are going to be presented. Their proofs, however, are going to be omitted as it is beyond the scope of this lesson.
Consider a circle, a chord, and a tangent to the circle through one of the endpoints of the chord. An alternate angle is an inscribed angle that subtends the given chord. The theorem states that the angle formed between the tangent and the chord is equal to the alternate angle. Figure 3 depicts this.
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Chord {eq}CD {/eq}, tangent that contains segment {eq}FG {/eq}, and alternate angle {eq}\angle CED {/eq} equal to {eq}\angle CDF {/eq}.
Two or more angles that subtend the same segment are equal, as angles {eq}\angle CED {/eq} and {eq}\angle CHD {/eq} in Figure 4.
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A circular sector is bounded by two radii and an arc, whereas a segment of a circle is enclosed by a chord and an arc. The area of a minor segment of a circle is {eq}A_{min~seg}~=~\frac{r^2x}{2}~-~\frac{r^2\sin x}{2} {/eq} and the corresponding major segment of the circle area is {eq}A_{maj~seg}~=~\pi r^2~-~\frac{r^2x}{2}~+~\frac{r^2\sin x}{2} {/eq}, provided that the radius is {eq}r {/eq} and the central angle {eq}x {/eq} is given in radians.
Another way of thinking is considering that the minor segment area is the area of a circular sector minus the area of a triangle, whereas the major segment area is the whole circle minus the minor segment area. It might be the case that the chord that determines two segments of a circle is the diameter, in which case the segments will be semi-circles, which have area {eq}\frac{\pi r^2}{2} {/eq}.
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