Area of a Segment of a Circle

Anderson Gomes Da Silva, Beverly Maitland-Frett
• Author
Anderson Gomes Da Silva

Anderson holds a Bachelor's and Master's Degrees (both in Mathematics) from the Fluminense Federal University and the Pontifical Catholic University of Rio de Janeiro, respectively. He was a Teaching Assistant at the University of Delaware (UD) for two and a half years, leading discussion and laboratory sessions of Calculus I, II and III. In the Winter of 2021 he was the sole instructor for one of the Calculus I sections at UD.

• Instructor
Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

Learn what a segment of a circle is. Discover the formula for finding the area of a segment of a circle, and identify theorems for segments of a circle. Updated: 11/16/2021

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What is a Segment of a Circle?

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.

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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Properties of Segments of a Circle

What 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.

The Segment of a Circle Formula

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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