11. AREAS RELATED TO CIRCLES

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.

A sector is a 'slice' of a circle, formed between two radii and the arc connecting them. For example, if you imagine cutting a pizza, each slice would represent a sector. A segment, on the other hand, is the area between a chord (a straight line connecting two points on the circle) and the arc connecting those two points. You can think of it as the area above a chord in a pizza slice.

Imagine a pie. When you cut two slices of the pie, each slice is a sector. If you have one slice that has the crust (arc) and the rest of the pie below that crust is a shape formed by a chord, that area is the segment.

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 a circle, a sector can be classified as either a minor sector or a major sector depending on the angle. The minor sector is the smaller section created by an angle less than 180°, while the major sector is larger, created by an angle greater than 180°. The sum of the angles in a circle is always 360°, so if you know the angle of the minor sector, you can easily find the angle of the major sector by subtracting that from 360°.

Think about a clock. If the hour hand points to 1 and the minute hand points to 12, the angle between them is the minor sector. If the hour hand moves to 11, the angle between these two hands now represents the major sector.

When degree measure of the angle at the centre is 360, area of the sector = pr2. So, 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 q, area of the sector = (πr² × q) / 360.

The area of a sector can be derived from the total area of the circle, which is πr², where r is the radius. Since a circle has 360 degrees, if you want to find the area for a smaller angle (θ degrees), you multiply the total area by the fraction of the angle out of 360. This gives you the area of the sector corresponding to that angle.

Imagine a garden shaped like a pie. If you want to determine the area of a slice that represents a quarter of the pie (90 degrees), you realize that this is one-fourth of the pie. So the calculation resembles taking a quarter of the total area of the circular garden.

Now a natural question arises: Can we find the length of the arc APB corresponding to this sector? Yes. 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.

To find the length of an arc corresponding to a sector, you use a similar approach as finding the area but with the circumference of the entire circle instead. The entire circumference is 2πr, and like the area, you take the fraction of the desired angle over 360 degrees to determine the length of the arc. This fraction shows how much of the whole circumference is represented by the sector.

If you've ever had a circular cake and you cut a slice out, the edge of that slice is the arc. If the cake's total edge (circumference) is based on its size, you would calculate the length of just the portion you cut off using proportions, like finding how much of a pizza crust you've eaten compared to the whole pizza.

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 Δ OAB.

The area of a segment of a circle is found by subtracting the area of the triangle formed by the two radii and the chord from the area of the sector. This means you're looking at only the curved portion of the sector that doesn't include the triangular 'point' at the center.

If you picture a pie slice again, the piece that sticks out toward the center (the triangle formed by the two radii) is not what you want to eat. So you subtract the area of that triangle from the area of the entire slice to find just the area of the pie that you can actually eat.

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