Explore the concepts of calculating areas related to circles, including sectors, segments, and combinations with other geometric shapes.
Areas Related to Circles - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Areas Related to Circles from Mathematics for Class X (Mathematics).
Basic comprehension exercises
Strengthen your understanding with fundamental questions about the chapter.
Questions
Define a sector and a segment of a circle. How do you calculate their areas?
Recall the definitions and formulas for sector and segment areas from the chapter.
Solution
A sector of a circle is the portion of the circular region enclosed by two radii and the corresponding arc. A segment is the portion of the circular region enclosed between a chord and the corresponding arc. The area of a sector with angle θ in degrees is given by (θ/360) × πr², where r is the radius. The area of a segment is found by subtracting the area of the triangle formed by the two radii and the chord from the area of the corresponding sector. For example, if a sector has a radius of 7 cm and an angle of 60°, its area is (60/360) × π × 7² ≈ 25.67 cm². The area of the corresponding segment would then depend on the area of the triangle formed.
How do you find the area of a quadrant of a circle? Provide a step-by-step solution for a circle with a circumference of 22 cm.
Use the circumference to find the radius first, then apply the sector area formula for 90°.
Solution
A quadrant is a sector with an angle of 90°. To find its area, first determine the radius of the circle. Given the circumference C = 2πr = 22 cm, solving for r gives r = 22 / (2π) ≈ 3.5 cm. The area of the quadrant is then (90/360) × π × r² = (1/4) × π × (3.5)² ≈ 9.62 cm². This method applies the sector area formula specifically for a 90° angle.
Explain how to calculate the length of an arc of a sector with a given angle and radius. Use an example with a radius of 14 cm and an angle of 30°.
The arc length is a fraction of the total circumference, based on the sector's angle.
Solution
The length of an arc (L) of a sector is calculated using the formula L = (θ/360) × 2πr, where θ is the angle in degrees and r is the radius. For a sector with r = 14 cm and θ = 30°, L = (30/360) × 2 × π × 14 ≈ (1/12) × 2 × 3.14 × 14 ≈ 7.33 cm. This formula derives from the proportion of the sector's angle to the full circle's angle (360°), applied to the total circumference (2πr).
Describe the process to find the area of a minor segment of a circle with radius 21 cm and a central angle of 120°.
Calculate the sector area first, then subtract the area of the triangle formed by the radii and chord.
Solution
To find the area of a minor segment, first calculate the area of the corresponding sector: (120/360) × π × 21² ≈ 462 cm². Then, find the area of the triangle formed by the two radii and the chord. Using trigonometry, the triangle's area is (1/2) × 21 × 21 × sin(120°) ≈ 190.53 cm². The segment's area is the sector's area minus the triangle's area: 462 - 190.53 ≈ 271.47 cm². This method combines geometric and trigonometric concepts.
What is the difference between a minor sector and a major sector? How would you calculate the area of the major sector if the minor sector's angle is 60° and the radius is 10 cm?
The major sector's angle is 360° minus the minor sector's angle. Use this in the sector area formula.
Solution
A minor sector has a smaller angle (less than 180°), while a major sector has a larger angle (more than 180°). For a minor sector with θ = 60° and r = 10 cm, the major sector's angle is 360° - 60° = 300°. The area of the major sector is (300/360) × π × 10² ≈ 261.67 cm². This approach uses the remaining angle after accounting for the minor sector's angle.
How can you determine the area between two consecutive ribs of an umbrella with 8 ribs and a radius of 45 cm?
Divide the full circle into 8 equal sectors and calculate one sector's area.
Solution
The area between two consecutive ribs is a sector of the circle formed by the umbrella. With 8 equally spaced ribs, each sector's angle is 360°/8 = 45°. The area of one sector is (45/360) × π × 45² ≈ 795.22 cm². This represents the area between two consecutive ribs, calculated using the sector area formula for the given angle and radius.
A horse is tied to a peg at the corner of a square field with a 5 m rope. Calculate the area of the field the horse can graze.
The grazing area is a sector with a 90° angle, as the field's corner limits the horse's movement.
Solution
The horse can graze a quarter-circle area with radius 5 m, as it's tied to a corner of the square field. The area of a full circle is π × 5² ≈ 78.54 m², so the quarter-circle area is (1/4) × 78.54 ≈ 19.64 m². This is the grazing area, limited by the rope's length and the field's boundary.
Explain how to find the total area cleaned by a car's wiper blade that sweeps a 115° angle with a length of 25 cm.
Calculate the area for one wiper's sweep and multiply by two for the total area.
Solution
The wiper blade sweeps a sector of a circle with radius 25 cm and angle 115°. The area cleaned is (115/360) × π × 25² ≈ 627.61 cm². Since there are two wipers, the total area cleaned is 2 × 627.61 ≈ 1255.22 cm². This calculation uses the sector area formula for each wiper's sweep.
A lighthouse warns ships with a light over an 80° sector to a distance of 16.5 km. Find the area of the sea warned.
Use the sector area formula with the given angle and radius (distance).
Solution
The warned area is a sector with radius 16.5 km and angle 80°. The area is (80/360) × π × (16.5)² ≈ 190.85 km². This represents the sea area over which ships are warned, calculated using the sector area formula with the given angle and distance as the radius.
Calculate the cost of making designs on a round table cover with six equal designs, each requiring an area calculation based on a radius of 28 cm, at a rate of ₹0.35 per cm².
Calculate one design's area as a 60° sector, then multiply by six and the cost rate.
Solution
First, find the area of one design. Since the cover has six equal designs, each design corresponds to a sector of 360°/6 = 60°. The area of one sector is (60/360) × π × 28² ≈ 410.67 cm². The total area for six designs is 6 × 410.67 ≈ 2464.02 cm². The cost is 2464.02 × ₹0.35 ≈ ₹862.41. This involves calculating each design's area as a sector and then scaling up for all designs.
Areas Related to Circles - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Areas Related to Circles to prepare for higher-weightage questions in Class X Mathematics.
Intermediate analysis exercises
Deepen your understanding with analytical questions about themes and characters.
Questions
A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.73)
Remember to convert the angle to radians if necessary for the sine function in the area of the triangle formula.
Solution
To find the areas of the minor and major segments, first calculate the area of the sector using the formula (θ/360) × πr². Then, find the area of the triangle formed by the two radii and the chord using the formula (1/2) × r² × sin(θ). Subtract the area of the triangle from the area of the sector to get the area of the minor segment. The area of the major segment is the area of the circle minus the area of the minor segment.
An umbrella has 8 ribs which are equally spaced. Assuming the umbrella to be a flat circle of radius 45 cm, find the area between two consecutive ribs of the umbrella.
Calculate the central angle for one sector by dividing the total angle of the circle by the number of ribs.
Solution
The area between two consecutive ribs is the area of a sector of the circle with angle 45° (since 360°/8 = 45°). Use the sector area formula (θ/360) × πr² to find this area.
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. Find the area of that part of the field in which the horse can graze.
Visualize the grazing area as a sector of a circle with a 90° angle because the field is square and the horse is tied at a corner.
Solution
The area grazed by the horse is a quarter-circle with radius equal to the length of the rope. Use the formula for the area of a circle (πr²) and divide by 4 since the horse is tied at the corner of the field.
Find the area of the segment AYB shown in a circle with radius 21 cm and angle AOB = 120°. (Use π = 22/7)
Use the sine of 120° which is √3/2 to calculate the area of the triangle.
Solution
First, find the area of the sector OAYB using (θ/360) × πr². Then, find the area of triangle OAB using (1/2) × r² × sin(θ). Subtract the area of the triangle from the area of the sector to get the area of the segment AYB.
A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades.
Ensure that the angle is in degrees when using the sector area formula.
Solution
Calculate the area swept by one wiper using the sector area formula (θ/360) × πr², then multiply by 2 since there are two wipers.
To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use π = 3.14)
Convert the radius to meters or kilometers consistently in your calculations.
Solution
The area warned is the area of a sector with radius 16.5 km and angle 80°. Use the sector area formula (θ/360) × πr² to find this area.
A round table cover has six equal designs as shown in a circle of radius 28 cm. Find the cost of making the designs at the rate of ₹0.35 per cm². (Use √3 = 1.7)
Each design is a sector of the circle with a 60° angle since the designs are equally spaced.
Solution
First, find the area of one design by dividing the area of the circle by 6. Then, calculate the total cost by multiplying the area of one design by 6 and then by the rate per cm².
A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find the total length of the silver wire required.
Remember that the length of one diameter is equal to the diameter of the circle.
Solution
Calculate the circumference of the circle using πd, then add the length of the 5 diameters to find the total length of the silver wire required.
Find the area of a quadrant of a circle whose circumference is 22 cm.
Use the value of π as 22/7 to simplify the calculations.
Solution
First, find the radius of the circle using the circumference formula C = 2πr. Then, calculate the area of the quadrant which is one-fourth of the area of the circle.
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
Calculate the angle swept by the minute hand per minute first, then multiply by the number of minutes.
Solution
The minute hand sweeps a sector with angle 30° in 5 minutes (since 360°/12 = 30° for each 5-minute interval). Use the sector area formula (θ/360) × πr² to find this area.
Areas Related to Circles - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Areas Related to Circles in Class X.
Advanced critical thinking
Test your mastery with complex questions that require critical analysis and reflection.
Questions
A circular park of radius 20 m is surrounded by a path of width 2 m. Find the area of the path. Also, find the cost of paving the path at the rate of ₹50 per m².
Consider the path as a ring around the park. The radius of the outer circle will be the sum of the park's radius and the path's width.
Solution
First, calculate the area of the larger circle (park + path) and subtract the area of the park. Then, multiply the area of the path by the cost per m².
An arc of a circle subtends an angle of 60° at the centre. If the length of the arc is 22 cm, find the radius of the circle.
The formula relating arc length, radius, and central angle is crucial here.
Solution
Use the formula for the length of an arc to find the radius. Remember to convert the angle to radians if necessary.
A sector of a circle with radius 14 cm has an area of 154 cm². Find the angle subtended at the centre by the sector.
The area of a sector is proportional to its central angle.
Solution
Apply the area of a sector formula to solve for the angle. Ensure the angle is in degrees.
In a circle of radius 21 cm, a chord subtends an angle of 120° at the centre. Find the area of the minor segment formed by the chord.
You'll need to find the area of the triangle using trigonometry.
Solution
Calculate the area of the sector and subtract the area of the triangle formed by the radii and the chord.
Two circles touch externally. The sum of their areas is 130π cm², and the distance between their centres is 14 cm. Find the radii of the two circles.
The distance between centres is the sum of the radii for externally touching circles.
Solution
Set up equations based on the sum of areas and the distance between centres. Solve the system of equations.
A horse is tied to a corner of a square field with a rope of length 7 m. Find the area the horse can graze. What if the rope was 14 m long?
Remember, the horse is tied to a corner, so the grazable area is a quarter-circle for the shorter rope.
Solution
The grazable area is a sector of a circle. Calculate the area for both rope lengths, considering the field's dimensions.
A circular wire of radius 42 cm is bent into the shape of a rectangle whose sides are in the ratio 6:5. Find the area of the rectangle.
The circumference of the circle equals the perimeter of the rectangle.
Solution
First, find the perimeter of the circle, which becomes the perimeter of the rectangle. Use the ratio to find the sides and then the area.
Find the area of the largest circle that can be inscribed in a square of side 14 cm.
Visualize the circle fitting perfectly inside the square, touching all four sides.
Solution
The diameter of the largest inscribed circle equals the side of the square. Use this to find the radius and then the area.
A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc.
The triangle formed by the two radii and the chord is equilateral if the chord equals the radius.
Solution
This forms an equilateral triangle with the two radii. Use properties of equilateral triangles to find the angle.
The minute hand of a clock is 14 cm long. Find the area swept by the minute hand in 10 minutes.
A minute hand completes 360° in 60 minutes, so in 10 minutes, it sweeps 60°.
Solution
Calculate the angle swept in 10 minutes and then use the area of a sector formula.
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