Hello students, welcome to today's mathematics lesson. I am so happy to be here with you to explore a very interesting and practical chapter – Chapter 11: Areas Related to Circles. You know, circles are all around us – from the wheels of a bicycle to the face of a clock, from the wheels of a car to the design of a bhelpuri plate! Understanding how to calculate areas related to circles will help you solve many real-life problems. So let's begin our journey together.
In this chapter, we are going to learn how to calculate the area of a sector and the area of a segment of a circle. These are two very important concepts that build upon what you already know about circles. So before we start, let me remind you of some basics.
You have already studied about circles in your earlier classes. You know that a circle is a set of points in a plane that are at a fixed distance from a center point. The distance from the center to any point on the circle is called the radius. The distance around the circle is called the circumference, and we calculate it using the formula 2πr, where r is the radius.
Now, students, let me ask you a question. Have you ever looked at a pizza and noticed how it can be cut into different slices? Each slice of pizza is actually a part of a circle, and it has a special name. In mathematics, we call such a slice a "sector" of the circle. Similarly, if you take a circular cake and cut out a piece, the remaining part or the piece itself represents another shape that we call a "segment" of the circle.
So let's understand these terms properly.
A sector of a circle is the portion or part of the circular region that is enclosed by two radii and the corresponding arc. Imagine you have a circle with center O. Now draw two radii OA and OB. The region bounded by these two radii and the arc AB is called a sector. The angle AOB is called the angle of the sector. Now, students, notice that there are actually two sectors here – one is the smaller region between the two radii, and the other is the larger region that goes around the rest of the circle. The smaller one is called the minor sector, and the larger one is called the major sector. The angle of the major sector would be 360 degrees minus the angle of the minor sector.
Similarly, a segment of a circle is the portion or part of the circular region that is enclosed between a chord and the corresponding arc. If you draw a chord AB in a circle, it divides the circle into two parts. The smaller part bounded by the chord AB and the arc AB is called the minor segment, and the larger part is called the major segment.
Now, students, an important point to remember is that whenever we write "segment" or "sector" in this chapter, we will generally mean the minor segment and the minor sector, unless stated otherwise. This is because when we talk about areas, we usually focus on the smaller portions.
Now that we have understood what sectors and segments are, let's learn how to calculate their areas. This is the main part of this chapter.
Let's start with finding the area of a sector. Consider a sector OAPB of a circle with center O and radius r. Let the degree measure of angle AOB be θ. We already know that the area of the entire circle is πr². Now, think about this – the entire circle can be considered as a sector that makes an angle of 360 degrees at the center. So, if a full circle (which is a sector of 360 degrees) has an area of πr², then we can use the unitary method to find the area of a sector with any other angle.
When the angle at the center is 360 degrees, the area of the sector is πr². So, when the angle is just 1 degree, the area would be πr² divided by 360. Therefore, when the angle is θ degrees, the area would be θ times that value. So we get:
Area of a sector of angle θ = (θ/360) × πr²
This is our first important formula, students. Remember that r is the radius of the circle and θ is the angle of the sector in degrees.
Now, a natural question arises – can we also find the length of the arc that forms the boundary of this sector? Yes, we can! Again, using the unitary method, we know that the entire circumference of the circle (which is the arc for a 360-degree sector) is 2πr. So, for a sector with angle θ, the length of the arc would be (θ/360) × 2πr.
So our second formula is: Length of an arc of a sector of angle θ = (θ/360) × 2πr.
Now, let's learn about finding the area of a segment. Consider a segment APB of a circle with center O and radius r. You can see that the segment APB is actually the region left after we remove the triangle OAB from the sector OAPB. In other words:
Area of segment APB = Area of sector OAPB – Area of triangle OAB
We already know how to find the area of the sector. For the triangle, we can use the formula (1/2) × base × height, or we can use trigonometry if needed.
Now, students, let me also tell you about the major sector and major segment. The area of the major sector would be the total area of the circle minus the area of the minor sector. Similarly, the area of the major segment would be the total area of the circle minus the area of the minor segment.
Now, let's look at some examples to understand these concepts better.
Example 1: Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major sector. Use π = 3.14.
Let's solve this step by step. We are given radius r = 4 cm and angle θ = 30 degrees. Using our formula:
Area of the sector = (θ/360) × πr² = (30/360) × 3.14 × 4 × 4 = (1/12) × 3.14 × 16 = (3.14 × 16)/12 = 50.24/12 = 4.19 cm² approximately.
Now, to find the area of the major sector, we subtract the area of the minor sector from the total area of the circle. The total area is πr² = 3.14 × 16 = 50.24 cm². So:
Area of major sector = 50.24 – 4.19 = 46.05 cm², which is approximately 46.1 cm².
Alternatively, we can also calculate the major sector directly using the formula (360 - θ)/360 × πr² = (330/360) × 3.14 × 16 = 46.05 cm².
So students, the key thing to remember is that the minor sector and major sector together make up the entire circle.
Now, let's look at another example which is a bit more complex.
Example 2: Find the area of the segment AYB shown in the figure, if radius of the circle is 21 cm and angle AOB is 120 degrees. Use π = 22/7.
This example is interesting because we need to find the area of a segment, which requires us to subtract the area of a triangle from the area of a sector.
First, let's understand what we need to find. The segment AYB is the region bounded by the chord AB and the arc AYB. This is the minor segment.
We know that: Area of segment AYB = Area of sector OAYB – Area of triangle OAB
Let's find each part.
First, area of sector OAYB. Using our formula: Area = (θ/360) × πr² = (120/360) × (22/7) × 21 × 21 = (1/3) × (22/7) × 441 = (22 × 441)/(21) = 9702/21 = 462 cm²
Now, to find the area of triangle OAB, we need to find its base and height. In this case, we can use trigonometry. Since OA = OB (both are radii), triangle OAB is isosceles. If we draw a perpendicular from O to the chord AB at point M, this perpendicular will bisect the chord AB and also bisect the angle AOB. So, angle AOM = angle BOM = half of 120° = 60°.
Now, in right triangle OMA, we have: OA = 21 cm (radius) Angle AOM = 60°
We can find OM (the height) using cosine: cos 60° = OM/OA 1/2 = OM/21 OM = 21/2 = 10.5 cm
Similarly, we can find AM using sine: sin 60° = AM/OA √3/2 = AM/21 AM = 21√3/2 cm
Since M is the midpoint of AB, the full length of AB = 2 × AM = 2 × (21√3/2) = 21√3 cm.
Now, area of triangle OAB = (1/2) × base × height = (1/2) × AB × OM = (1/2) × 21√3 × 21/2 = (21 × 21 × √3)/(4) = 441√3/4 cm²
Now, we can find the area of the segment: Area of segment AYB = Area of sector – Area of triangle = 462 – 441√3/4 = (1848/4 – 441√3/4) = (1848 – 441√3)/4 = 21/4 (88 – 21√3) cm²
This is the exact answer. If we use √3 = 1.73, we can get a numerical approximation.
Students, this example shows us an important technique. When we need to find the area of a segment, we often need to find the area of the triangle formed by the two radii and the chord. This requires us to use trigonometry to find the height and base of the triangle.
Now, let's summarize what we have learned in this chapter so far.
First, we learned about sectors and segments of a circle. A sector is the region bounded by two radii and an arc, while a segment is the region bounded by a chord and an arc. Each of these has a minor and major version.
Then we derived the formula for the area of a sector: Area = (θ/360) × πr², where θ is the angle in degrees and r is the radius.
We also learned that the length of an arc is (θ/360) × 2πr.
For finding the area of a segment, we use: Area of segment = Area of sector – Area of the triangle formed by the two radii and the chord.
These are the fundamental concepts of this chapter. Understanding these formulas and knowing how to apply them will help you solve many problems related to circles.
In our next lessons, we will practice more problems to strengthen your understanding. But for now, make sure you remember these formulas and understand how they are derived.
Remember, mathematics is not just about memorizing formulas – it's about understanding the logic behind them. So always try to understand why a formula works, and you will be able to apply it more effectively.
That's all for today, students. Keep practicing, and I will see you in the next lesson. Goodbye!
Now, let me give you a complete summary of everything we have learned in this chapter:
In this chapter on Areas Related to Circles, we studied several important concepts. First, we recalled the definitions of sector and segment of a circle. A sector is the portion of a circle enclosed by two radii and the corresponding arc, while a segment is the portion enclosed between a chord and the corresponding arc. Each has a minor and major version, with the minor having the smaller angle or area.
We derived the formula for the length of an arc of a sector: it is given by θ/360 multiplied by 2πr, where r is the radius and θ is the angle in degrees.
We derived the formula for the area of a sector: it is given by θ/360 multiplied by πr². This comes from considering the entire circle as a sector of 360 degrees and applying the unitary method.
We learned that the area of a segment of a circle is equal to the area of the corresponding sector minus the area of the corresponding triangle formed by the two radii and the chord.
We worked through two examples to understand these concepts better. The first example showed us how to find the area of a sector and then use it to find the area of the major sector. The second, more complex example showed us how to find the area of a segment by calculating the area of the sector and then subtracting the area of the triangle, where we used trigonometry to find the dimensions of the triangle.
These concepts have many practical applications in real life, such as calculating the area of a slice of pizza, the area of a piece of a circular cake, or any situation where we need to find the area of a part of a circular region.
That concludes our lesson on Chapter 11: Areas Related to Circles. Thank you for your attention, students. Keep practicing and never hesitate to ask questions if you have any doubts. Good luck with your studies!