Areas Related to Circles — Class 10 Maths Notes

Topics covered in Areas Related to Circles

1. 1.Introduction of Area Related to Circle

   Let's understand the topic of "Areas Related to Circles" in a fun and simple way. This topic is quite interesting and useful in both mathematics and in real life.

   Basic Concept: A circle is a round shape, like a coin or a dinner plate. The distance around the circle is called the circumference, and the line from the center of the circle to any point on its edge is the radius.

   1. Area of a Circle: The area is the space inside the circle. We calculate it using the formula: Area=×radius2Area=π×radius2 Here, π (pi) is approximately 3.14. So, if the radius is 5 cm, the area would be 3.14×5×5=78.5 cm23.14×5×5=78.5 cm2.

   2. Circumference of a Circle: The circumference is the distance around the circle, calculated using: Circumference=2××radiusCircumference=2×π×radius So, with a radius of 5 cm, the circumference is 2×3.14×5=31.4 cm2×3.14×5=31.4 cm.

   3. Areas of Sectors and Segments: Imagine a pizza slice. This slice is part of a circle, called a sector. The area of a sector is a fraction of the circle's area, depending on the angle of the slice. A segment is like a sector but with the triangle part cut off.

   Real-Life Examples and Applications:

   • Pizza Slices: When you cut a pizza into slices, you are dividing the circle into sectors. Understanding the area of each slice helps in dividing the pizza equally.
   • Wheel Design: The design of wheels in cars and bikes involves calculations of circumference and area to ensure they fit properly and function well.
   • Architecture and Engineering: Architects and engineers use these concepts to design circular structures, like roundabouts or domed roofs.

   Careers and Industries:

   • Mathematics and Engineering: These concepts are fundamental in fields like mechanical engineering, architecture, and design.
   • Education: As a teacher or academician, understanding and explaining these concepts is vital.

2. 2.Areas of Sector and Segment of a Circle

   Let's explore the areas of sectors and segments of a circle, which are quite fascinating parts of circle geometry.

   Sector of a Circle: A sector is like a 'piece of pie' or 'pizza slice' in a circle. It is formed by two radii and the arc between them.

   • Formula to Calculate Area of a Sector: Area of Sector = (θ / 360°) × π × r² Here, θ is the central angle in degrees, and r is the radius of the circle.

     Example: If a sector has a central angle of 90° and the radius of the circle is 10 cm, the area of the sector is: (90° / 360°) × π × 10² = ¼ × π × 100 = 25π cm²

   Segment of a Circle: A segment is a region in a circle separated by a chord (a line that links two points on the circle's edge). It's like a sector but minus the triangular part.

   • Formula to Calculate Area of a Segment: Area of Segment = Area of Sector - Area of Triangle The area of the triangle can be found using various methods, depending on the given information.

     Example: For a segment with the same dimensions as the previous sector example, if the height of the corresponding triangle is 6 cm, the area of the triangle is ½ × base × height. The base here is the chord length, which can be calculated using circle geometry principles.

   Real-Life Examples and Careers:

   • Design and Architecture: Understanding these areas is essential in designing elements like arches or circular windows.
   • Agriculture: Farmers might use these calculations to determine areas for irrigation or planting in circular fields.
   • Sports: Planning athletic tracks or fields often involves calculations of sectors and segments.

   Activity:

   • Paper Activity: Cut out a circle from paper, and then cut a 'slice' to create a sector. Measure the angle and radius, and try calculating the area of the paper sector.
   • Field Activity: On a playground, try marking a sector or segment on the ground using chalk. Measure and calculate its area.

3. 3.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).

   To find the area of a sector of a circle with radius 4 cm and a central angle of 30°, we can use the following formula:

   Area of Sector = (θ / 360°) × π × r²

   Here: θ = 30° (the central angle) r = 4 cm (the radius) π = 3.14 (the value of pi as given)

   Let's calculate the area of the sector.

   Area of Sector = (30° / 360°) × 3.14 × (4 cm)²

   Now, to find the area of the corresponding major sector (which is the larger part of the circle remaining after the minor sector is removed), we subtract the area of the minor sector from the total area of the circle.

   The total area of the circle is given by the formula:

   Area of Circle = π × r²

   We can calculate the area of the major sector as follows:

   Area of Major Sector = Area of Circle - Area of Minor Sector

   Let's calculate these areas step by step.

   The area of the minor sector (the smaller section of the circle corresponding to a 30° angle) with a radius of 4 cm is approximately 4.19 cm².

   The area of the corresponding major sector (the larger part of the circle) is approximately 46.05 cm².

4. 4.Find the area of a quadrant of a circle whose circumference is 22 cm.

   To find the area of a quadrant of a circle, we first need to know the radius of the circle. A quadrant is one-fourth of a circle, so once we have the radius, we can calculate the area of the entire circle and then divide by 4 to get the area of the quadrant.
   

   First, let's find the radius using the circumference. The circumference (C) of a circle is given by the formula:

   =2C=2πr

   Given that the circumference is 22 cm, we can solve for the radius (r):

   22=222=2πr

   To find r, we divide both sides by 22π:

   =222r=2π22​

   Once we have the radius, we can find the area of the entire circle (A) using the formula:

   =2A=πr2

   And the area of the quadrant (A_q) will be:

   =4Aq​=4A​

   Let's calculate this step by step, using π as 3.14 for the calculation.

   The radius of the circle with a circumference of 22 cm is approximately 3.50 cm.

   The area of a quadrant of this circle is approximately 9.63 cm².

5. 5.The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

   
   

   To find the area swept by the minute hand in 5 minutes, we need to calculate the sector of the circle that the minute hand covers as it moves around the clock.

   Here's how we can calculate it:

   1. Calculate the angle swept by the minute hand in 5 minutes. Since the minute hand completes a full rotation (360 degrees) in 60 minutes, in 5 minutes it would sweep: Angle=360 degrees60 minutes×5 minutes
      Angle=60 minutes360 degrees​×5 minutes

   2. Use the sector area formula to find the area swept by the minute hand: Area of Sector=Angle360××2
      Area of Sector=360Angle​×π×r2 where r is the length of the minute hand and π is approximately 3.14159.

   Let's perform the calculation.

   The area swept by the minute hand of a clock, which is 14 cm long, in 5 minutes is approximately 51.31 cm². ​

6. 6.A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use π = 3.14)

   To solve this problem, we'll first find the area of the minor segment and then the area of the major sector of a circle with a radius of 10 cm, where the chord subtends a right angle at the center.
   

   1. Area of the Minor Segment:

      • The minor segment is the smaller part of the circle cut off by the chord. To find its area, we first find the area of the sector formed by the 90° angle and then subtract the area of the triangle formed by the radius and the chord.

      • Area of the Sector: The angle is 90° (right angle). So, the sector's area is 90°360°××2360°90°​×π×r2.

      • Area of the Triangle: In a right-angled triangle formed by the radii and the chord, the sides (radii) are both 10 cm (length of the radius). The area of a right triangle is 12×base×height21​×base×height.

      • The area of the minor segment = Area of the sector - Area of the triangle.

   2. Area of the Major Sector:

      • The major sector is the larger part of the circle. Its area is the total area of the circle minus the area of the minor sector.

      • Total Area of the Circle: ×2π×r2.

      • The area of the major sector = Total area of the circle - Area of the minor sector.

   Now, let's calculate these areas using =3.14π=3.14 and =10r=10 cm.

   The area of the minor segment of the circle is approximately 28.5 cm², and the area of the major sector is approximately 235.5 cm².

   Here's a quick summary of the steps:

   1. The area of the 90° sector was calculated as 90°360°××2=78.5360°90°​×π×r2=78.5 cm².
   2. The area of the right-angled triangle formed by the radii and the chord was 12×base×height=5021​×base×height=50 cm².
   3. The area of the minor segment was found by subtracting the area of the triangle from the area of the sector: 78.5−50=28.578.5−50=28.5 cm².
   4. The area of the major sector was the total area of the circle minus the area of the 90° sector: 314−78.5=235.5314−78.5=235.5 cm².

7. 7.In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc (iii) area of the segment formed by the corresponding chord

   To solve this problem, we need to use the properties of a circle and the relationships between angles, arc lengths, and areas of sectors and segments.

   1. Length of the Arc: The length of an arc in a circle is given by the formula =360×2L=360θ​×2πr, where θ is the central angle in degrees, r is the radius, and π is approximately 3.14.

      Given:

      • =60°θ=60°
      • =21r=21 cm

      So, the length of the arc (L) is: =60°360°×2×3.14×21L=360°60°​×2×3.14×21

   2. The length of the arc is approximately 21.9921.99 cm. Area of the Sector: The area of a sector of a circle is given by the formula =360×2A=360θ​×πr2.

      Using the same values for θ and r, the area of the sector (A) is: =60°360°×3.14×212A=360°60°​×3.14×212

   3. The area of the sector formed by the 60° arc in a circle with radius 21 cm is approximately 230.91230.91 cm². Area of the Segment: The area of the segment is the area of the sector minus the area of the triangle formed by the radius and the chord.

      First, we calculate the area of the triangle. For a 60° segment in a circle, the triangle is equilateral. The area of an equilateral triangle with side s is given by 34×243​​×s2. The side length s of the triangle is the chord, which can be calculated using the formula for the chord length in a circle: =2sin⁡(2)s=2rsin(2θ​).

      So, the area of the triangle (T) is: =34×(2×21×sin⁡(30°))2T=43​​×(2×21×sin(30°))2

      Finally, the area of the segment is: Area of Segment=−
      Area of Segment=A−T
      The area of the segment formed by the corresponding chord is approximately is 39.95 cm².

8. 8.A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig). Find (i) the area of that part of the field in which the horse can graze. (ii) the increase in the grazing area if the rope were 10 m

   To find the grazing area for the horse and the increase in the grazing area if the rope were longer, we will calculate the area of the quarter circle that the horse can graze with ropes of 5 m and 10 m in length respectively, since the horse is tied to a corner of a square field.

   (i) Area of the grazing field with a 5 m rope: The horse can graze in the area of a sector of a circle with a 90° angle (which is a quarter of a circle) and a radius equal to the length of the rope.

   The area of the sector (quarter circle) is given by: Area=360××2Area=360θ​×π×r2 For a quarter circle, =90°θ=90°, =5r=5 m.

   Area5=90360××52Area5m​=36090​×π×52

   (ii) Increase in the grazing area if the rope were 10 m: The increased area is the difference between the areas of a quarter circle with a 10 m rope and the original 5 m rope.

   For a 10 m rope: Area10=90360××102Area10m​=36090​×π×102

   The increase in area is: Increase=Area10−Area5Increase=Area10m​−Area5m​

   Let's calculate these two areas using =3.14π=3.14.

   Here's the detailed solution with the calculated areas:

   (i) Area of the grazing field with a 5 m rope: Area5=90360×3.14×52Area5m​=36090​×3.14×52 Area5=14×3.14×25Area5m​=41​×3.14×25 Area5=19.625 m2Area5m​=19.625m2

   So, with a 5 m rope, the horse can graze an area of 19.625 m².

   (ii) Increase in the grazing area if the rope were 10 m long: Area10=90360×3.14×102Area10m​=36090​×3.14×102 Area10=14×3.14×100Area10m​=41​×3.14×100 Area10=78.5 m2Area10m​=78.5m2

   The increased area when the rope is extended to 10 m is: Increase=78.5 m2−19.625 m2Increase=78.5m2−19.625m2 Increase=58.875 m2Increase=58.875m2

   Therefore, if the rope were 10 m long instead of 5 m, the grazing area would increase by 58.875 m²

9. 9.An umbrella has 8 ribs which are equally spaced (see Fig.). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

   To find the area between two consecutive ribs of the umbrella, we need to calculate the area of the sector formed by two consecutive ribs.

   Given that the umbrella has 8 equally spaced ribs, it means the umbrella can be divided into 8 equal sectors. Each sector then subtends an angle of 360°8=45°8360°​=45° at the center of the circle that represents the umbrella.

   Here's how we can calculate the area of one of these sectors, which is the area between two consecutive ribs:

   1. Find the central angle for the sector between two ribs: Since there are 8 ribs, the central angle θ for one sector is 360°8=45°8360°​=45°.

   2. Calculate the area of the sector (which is the area between two ribs): The area A of a sector is given by the formula =360××2A=360θ​×π×r2, where r is the radius of the umbrella.

      Using the given radius of 45 cm, the area A between two ribs is: =45°360°×3.14×(45 cm)2A=360°45°​×3.14×(45 cm)2

   3. Perform the calculation: Now we'll compute the area using the values provided.

   Let's carry out the calculation.

   The area between two consecutive ribs of the umbrella is 794.8125794.8125 cm².

   Here's a detailed explanation of the calculation:

   1. The umbrella is divided into 8 equal sectors because it has 8 ribs.
   2. Each sector has a central angle of 360°8=45°8360°​=45°.
   3. The formula to calculate the area of a sector is =360××2A=360θ​×π×r2, where θ is the angle in degrees and r is the radius in cm.
   4. By inserting the values into the formula, we get: =45360×3.14×452A=36045​×3.14×452 =18×3.14×2025A=81​×3.14×2025 =0.125×3.14×2025A=0.125×3.14×2025 =794.8125 cm2A=794.8125 cm2

   Therefore, the area of the umbrella that one rib covers, or the area between two consecutive ribs, is 794.8125794.8125 cm²

10. 10.Quick Revision

    Area Related to Circles

    The area related to circles mainly involves finding the space inside a circle and parts of a circle. The key formula here is the area of a circle, which is Area=2Area=πr2, where r is the radius of the circle.

    Areas of Sector and Segment of a Circle

    • Sector: A sector is like a 'pizza slice' of the circle. The area of a sector is calculated by Area=360×2Area=360θ​×πr2, where θ is the angle of the sector.
    • Segment: A segment is a region between a chord and the corresponding arc. To find its area, first find the area of the sector and then subtract the area of the triangle formed by the chord and the two radii.

    Implementing Formulas

    1. 1. Identify what you need to find: the area of the whole circle, a sector, or a segment.
    2. 2. For a circle, use 2πr2.
    3. 3. For a sector, calculate the fraction of the circle using the angle, and then multiply by the area of the circle.
    4. 4. For a segment, subtract the area of the triangle from the area of the sector.