How Can You Find the Area Of The Major Segment Using The Minor Segment?

A segment is a part of a circle. Let us refresh few important terminologies related to a circle, to develop a better understanding of other parameters associated with a circle. A circle is a two-dimensional shape that consists of all points, located in the same plane such that each of these points is at a fixed distance when measured from a given point. That fixed point from which this measurement is done is called as the center of the circle. A circle can also be defined as a curve, which is created by a point, moving in a plane at a fixed distance from the reference point, lying inside this curve. This fixed distance, measured from the center is called as the radius of the circle. A circle is represented as shown in the figure below.

By referring to this figure above, a few of the basic terms associated with a circle can be understood easily. The center of a circle is usually denoted by the letter ‘O’. The fixed distance from the center of the circle to the periphery is the radius ‘R’ or ‘r’. The diameter of the circle is defined as a line segment that connects two points on the periphery of the circle, and this line must cross through the center of the circle. This is measured as twice the length of the radius of the circle and it is represented as D or ‘d’. The total length of the periphery of the circle or the boundary of a circle is called the Circumference and the total area enclosed by the boundary of the circle is called the area inside the circle. Thus, if the radius of the circle is measured as ‘r’ units, then the diameter of the circle will be of length 2r.

The circumference of the circle can be calculated as 2πr and the area enclosed by the circle is calculated by the formula. If a line joins any two points on the circle, such that the line does not pass through the center, then this line is called a chord. Diameter can be considered as a special type of chord which divides the circle in exactly half and passes through the center. A segment of a circle is the area, which is circumscribed by an arc and a chord of the circle. An arc of a circle is defined as a section of the circumference of the circle. Segments can be classified into two types.

One is called as the minor segment (represented in figure 2 as green area) and the other is called major segment (represented in figure 2 as the white unshaded portion in the circle). From the figure, it is clear that a minor segment is created by the minor arc and the major segment is created by a major arc of the circle. At the center of the circle, the total angle is 360 degrees. If Ɵ is the angle subtended by the minor segment at the center of the circle, then the angle subtended by the major segment or major arc at the center will be (360 – Ɵ) degrees.

Figure 2: Circle with minor and major segments, with usual notations.

From the figure, the sector is defined as the area swept by the two radius and the arc. As explained above, the total area of a circle is given by. This corresponds to 360 degrees. Thus, if we know the radius, we can determine the area. Once the area is calculated, we can determine the area of the sector by unitary method, to yield the following formula.

Now that the area of the minor sector is known, then from here, we can determine the area of the minor segment as well. If one looks closely, the sector can be assumed to be comprised of a triangle and a minor segment. The area of the triangle can be calculated with the help of the radius R.

For the triangle, the height, shown in the figure as ‘d’ will be and the base will be of length 2.r.sin⁡〖θ/2〗. These relations can be obtained from the properties of trigonometry.

On entering the values in the formula of a triangle, the area of the triangle can be calculated to be as follows,

Therefore, the area of the minor segment is given as Aminor.

As the area of the minor segment is now known, thus the area of the major segment can be easily calculated by subtracting the area of the minor segment from the area bounded by the circle. Thus, the formula for the area of major segment, represented as Amajor is as below.

Learn More: Areas Related To Circles From Class 10 Maths

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