11.1. Areas of Sector and Segment of a Circle

You have already come across the terms sector and segment of a circle in your earlier classes. Recall that the portion (or part) of the circular region enclosed by two radii and the corresponding arc is called a sector of the circle and the portion (or part) of the circular region enclosed between a chord and the corresponding arc is called a segment of the circle. Thus, in Fig. 11.1, shaded region OAPB is a sector of the circle with centre O. ∠ AOB is called the angle of the sector. Note that in this figure, unshaded region OAQB is also a sector of the circle. For obvious reasons, OAPB is called the minor sector and OAQB is called the major sector. You can also see that angle of the major sector is 360° – ∠ AOB.

In this chunk, we learn about two important parts of circles: sectors and segments. A sector is like a slice of pizza; it's the area enclosed by two radii (the edges of the slice) and the arc (the crust). A segment is more like a piece of a pizza that is restricted by a straight line (the chord) and the arc above that line. In the context of this definition, 'minor' refers to the smaller part (minor sector/segment) while 'major' refers to the larger part (major sector/segment). Understanding these definitions sets the groundwork for calculating areas.

Think of a birthday cake. If you take a slice (that’s the sector), it consists of the crust and the pieces of the cake (the area between the two radii). If you take a piece that’s above a partition line and below the crust (like some toppings), that’s a segment. It’s helpful to understand these concepts with everyday objects like cakes, pizzas, or even segments of fruit.

Now with this knowledge, let us try to find some relations (or formulae) to calculate their areas. Let OAPB be a sector of a circle with centre O and radius r (see Fig. 11.3). Let the degree measure of ∠AOB be θ. You know that area of a circle (in fact of a circular region or disc) is πr². In a way, we can consider this circular region to be a sector forming an angle of 360° (i.e., of degree measure 360) at the centre O. Now by applying the Unitary Method, we can arrive at the area of the sector OAPB as follows: When degree measure of the angle at the centre is 360, area of the sector = πr². When the degree measure of the angle at the centre is 1, area of the sector = πr²/360. Therefore, when the degree measure of the angle at the centre is θ, area of the sector = (θ/360) * πr². Thus, we obtain the following relation (or formula) for area of a sector of a circle: Area of the sector of angle θ = (θ/360) * πr².

Here, we derive the formula to calculate the area of a sector of a circle. The full circle has an area of πr², proportional to the angle subtended at the center. To find the area for a sector with an angle θ instead of 360°, we multiply the full area by the fraction of θ/360. This gives us a reliable way to calculate the area of any sector as long as we know the radius and the angle.

Imagine you have a pie that represents a full circle and it can be cut into 360 slices. If you only take one slice (1 degree), it would represent a tiny fraction of the total area. Now, if you take 30 slices (30 degrees), to find out how much area that is, you'd need to calculate the fraction of the total pie that your 30 slices represent. This is exactly how the formula works!

Now a natural question arises: Can we find the length of the arc APB corresponding to this sector? Yes. Again, by applying the Unitary Method and taking the whole length of the circle (of angle 360°) as 2πr, we can obtain the required length of the arc APB as (θ/360) * 2πr.

Similar to how we found the area of the sector, we can find the length of the arc relating to a specific angle. The entire circumference of a circle is 2πr, but if we're looking for just a segment of that, we again apply the fraction represented by θ/360 to find the length of the arc for any angle θ.

Think of a circular race track. If the entire length around the track (full circle) is the distance all the way around (2πr), then if you only run 90 degrees (a quarter of the race), the distance you ran is a quarter of the whole length. This same principle applies to calculating arc lengths; you merely multiply the total length by the fraction you've covered.

Now let us take the case of the area of the segment APB of a circle with centre O and radius r. You can see that: Area of the segment APB = Area of the sector OAPB – Area of ΔOAB = (θ/360) * πr² – Area of ΔOAB. Note: From Fig. 11.3 and Fig. 11.4 respectively, you can observe the Area of the major sector OAQB = πr² – Area of the minor sector OAPB and Area of major segment AQB = πr² – Area of the minor segment APB.

The area of the segment is derived from subtracting the area of the triangle formed by the radii and the chord from the area of the sector. Essentially, it tells us how much area is enclosed between the arc and the chord, highlighting the importance of both components in calculating areas of segments.

Let's use the pizza analogy again. If you have a slice of pizza (the sector) and you remove the triangular part of that slice that formed by the crust and the two cuts (the triangle), what remains is the segment, the part that is delicious but lacks the simple triangle shape. This analogy helps visualize how segments relate to sectors.

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