CBSE Class 10 Mathematics (2026–27)
Chapter 11: Areas Related to Circles
20 Important Questions and Answers
Q1. What is the difference between the circumference and area of a circle?
Answer:
The circumference of a circle is the length of its boundary, while the area is the region enclosed by the boundary. Circumference is measured in linear units and is calculated using the formula (C = 2\pi r), where (r) is the radius. Area is measured in square units and is calculated using (A = \pi r^2). For example, if the radius of a circle is 7 cm, its circumference is (44) cm and its area is (154) cm². Thus, circumference tells us how far around the circle is, whereas area tells us how much space is inside the circle.
Q2. Define a sector of a circle. How is its area calculated?
Answer:
A sector is the region enclosed by two radii and the corresponding arc of a circle. It resembles a slice of pizza. The area of a sector depends on the angle subtended at the centre. If the central angle is (\theta) degrees and the radius is (r), then the area of the sector is:
[\text{Area of Sector} = \frac{\theta}{360^\circ}\times \pi r^2]
For example, if the radius is 14 cm and the angle is 90°, the area of the sector is:
[\frac{90}{360}\times \pi \times 14^2 = 154 \text{ cm}^2]
Thus, a sector is a fractional part of the entire circle.
Q3. What is an arc? Differentiate between major and minor arcs.
Answer:
An arc is a portion of the circumference of a circle. When two points are marked on a circle, they divide the circumference into two arcs. The shorter arc is called the minor arc, while the longer arc is called the major arc. The measure of the minor arc is less than 180°, whereas the major arc measures more than 180°. Arc length is calculated using:
[\text{Arc Length}=\frac{\theta}{360^\circ}\times 2\pi r]
Arcs are important in determining the length and area of sectors. They are commonly used in geometry and practical applications such as designing circular tracks and wheels.
Q4. Explain the concept of a segment of a circle.
Answer:
A segment of a circle is the region enclosed by a chord and the arc corresponding to that chord. Segments are of two types: major segment and minor segment. The smaller region formed is called the minor segment, while the larger region is the major segment. The area of a segment can be found by subtracting the area of the triangle formed by the radii and chord from the area of the corresponding sector:
[\text{Area of Segment} = \text{Area of Sector} – \text{Area of Triangle}]
Segments help in solving practical problems involving curved boundaries and are an important topic in areas related to circles.
Q5. How do you find the length of an arc of a circle?
Answer:
The length of an arc is the distance along the curved boundary of a circle between two points. It depends on the radius and the central angle. The formula for finding arc length is:
[\text{Arc Length}=\frac{\theta}{360^\circ}\times 2\pi r]
where (\theta) is the central angle and (r) is the radius. For example, if (r = 7) cm and (\theta = 90^\circ),
[\text{Arc Length}=\frac{90}{360}\times 2\times \frac{22}{7}\times 7=11 \text{ cm}]
Arc length is useful in calculating distances along curved paths, such as roads and tracks.
Q6. What is the area of a semicircle and how is it calculated?
Answer:
A semicircle is half of a circle formed when a diameter divides the circle into two equal parts. Since it is half of the entire circle, its area is half the area of the circle.
[\text{Area of Semicircle}=\frac{1}{2}\pi r^2]
For example, if the radius is 14 cm:
[\frac{1}{2}\times \frac{22}{7}\times 14\times 14 = 308 \text{ cm}^2]
The perimeter of a semicircle includes the curved part and the diameter. Semicircles are often seen in architecture, bridges, windows, and various engineering designs.
Q7. Explain the formula for the area of a quadrant.
Answer:
A quadrant is one-fourth of a circle. It is formed when two perpendicular radii divide the circle into four equal parts. Since a quadrant represents one-fourth of the entire circle, its area is:
[\text{Area of Quadrant}=\frac{1}{4}\pi r^2]
For a circle of radius 14 cm:
[\frac{1}{4}\times \frac{22}{7}\times 14\times14 = 154 \text{ cm}^2]
Quadrants are useful in geometry problems and appear in practical structures such as quarter-circle gardens and decorative designs.
Q8. Why is π important in finding areas related to circles?
Answer:
The symbol π (pi) represents the ratio of the circumference of a circle to its diameter. Its approximate value is (3.14) or (\frac{22}{7}). Since all circles have curved boundaries, calculations involving their circumference, area, sectors, and segments require π. The formulas (C = 2\pi r) and (A = \pi r^2) are based on π. Without π, accurate measurements of circular regions would not be possible. It is a universal mathematical constant used extensively in geometry, engineering, physics, and architecture whenever circular shapes are involved.
Q9. What is meant by the perimeter of a sector?
Answer:
The perimeter of a sector is the total length of its boundary. A sector is bounded by two radii and an arc. Therefore, its perimeter is the sum of the lengths of the two radii and the arc length.
[\text{Perimeter of Sector}=2r+\frac{\theta}{360^\circ}\times 2\pi r]
For example, if (r = 7) cm and (\theta = 90^\circ), the arc length is 11 cm, so:
[\text{Perimeter}=14+11=25 \text{ cm}]
This concept is useful in finding fencing lengths and boundaries of circular sections.
Q10. Define concentric circles. How do they help in area calculations?
Answer:
Concentric circles are circles having the same centre but different radii. The region between two concentric circles is called an annulus or circular ring. To find the area of this ring, subtract the area of the smaller circle from the area of the larger circle:
[\text{Area of Ring}=\pi(R^2-r^2)]
where (R) is the larger radius and (r) is the smaller radius. Concentric circles are commonly seen in targets, wheels, and circular tracks. Their area calculations help determine the space occupied between two circular boundaries.
Q11. How is the area of a segment found?
Answer:
The area of a segment is obtained by subtracting the area of the triangle formed by the radii and chord from the area of the corresponding sector. The formula is:
[\text{Area of Segment} = \text{Area of Sector} – \text{Area of Triangle}]
This method is used because a segment consists of the sector minus the triangular portion. The area of the sector is calculated using the central angle, while the area of the triangle depends on its dimensions. Segment calculations are important in engineering, architecture, and land measurement where curved regions need to be measured accurately.
Q12. What is the significance of sectors in daily life?
Answer:
Sectors are commonly found in everyday life. Pizza slices, pie charts, cake pieces, and fan blades are examples of sectors. The concept helps in calculating the area and boundary of such objects. In statistics, pie charts use sectors to represent data proportionally. Engineers and architects use sectors while designing circular structures and machines. Understanding sectors also helps in solving practical problems involving circular parks, fountains, and tracks. Thus, sectors connect mathematical concepts with real-world applications and make geometry more meaningful and useful.
Q13. How do you calculate the area of a sector with a given central angle?
Answer:
The area of a sector depends on the central angle and the radius of the circle. Since a sector is a fraction of the whole circle, its area is calculated by multiplying the total area of the circle by the fraction (\frac{\theta}{360^\circ}).
[\text{Area of Sector}=\frac{\theta}{360^\circ}\times \pi r^2]
For example, if the radius is 10 cm and the angle is 72°:
[\frac{72}{360}\times \pi \times 10^2 = 20\pi]
cm². This formula helps determine the exact area occupied by a sector within a circle.
Q14. Explain the relationship between a sector and a circle.
Answer:
A sector is a part of a circle enclosed by two radii and the corresponding arc. Just as a slice is a part of a cake, a sector is a portion of the entire circle. The area and arc length of a sector depend on the central angle. If the angle is 360°, the sector becomes the complete circle. Thus, a sector represents a fraction of the whole circle. Understanding this relationship helps students solve problems involving arc lengths, sector areas, and practical applications such as pie charts and circular designs.
Q15. Why are areas related to circles important in geometry?
Answer:
Areas related to circles help us measure curved regions accurately. Concepts such as sectors, segments, semicircles, quadrants, and circular rings are used in solving practical geometry problems. These calculations are useful in engineering, architecture, construction, sports field design, and manufacturing. Understanding circular areas improves problem-solving skills and develops logical thinking. Since many natural and man-made objects have circular shapes, these concepts help connect mathematics with real-life situations. Therefore, the study of areas related to circles is an important part of geometry and everyday applications.
Q16. What is a circular ring and how is its area calculated?
Answer:
A circular ring, also known as an annulus, is the region between two concentric circles. It resembles the shape of a ring or washer. The area of the ring is found by subtracting the area of the smaller circle from the area of the larger circle.
[\text{Area of Ring}=\pi(R^2-r^2)]
where (R) is the outer radius and (r) is the inner radius. This concept is useful in designing wheels, tracks, pipes, and circular pathways. It helps determine the space enclosed between two circular boundaries.
Q17. How is the perimeter of a semicircle different from its area?
Answer:
The area of a semicircle measures the space enclosed by it, while the perimeter measures the total boundary length. The area is calculated using:
[\text{Area}=\frac{1}{2}\pi r^2]
The perimeter includes the curved arc and the diameter:
[\text{Perimeter}=\pi r + 2r]
Thus, area is expressed in square units, whereas perimeter is expressed in linear units. Understanding this difference is important because some problems require finding the enclosed region, while others require finding the length around the figure.
Q18. What role does the radius play in circle-related calculations?
Answer:
The radius is one of the most important measurements of a circle. It is the distance from the centre to any point on the circumference. Almost every formula related to circles depends on the radius. The circumference is (2\pi r), the area is (\pi r^2), and formulas for sectors, segments, and arcs also involve the radius. A small change in radius significantly affects the area because the radius is squared in the area formula. Therefore, accurate measurement of the radius is essential for obtaining correct results in circle-related calculations.
Q19. What is the difference between a chord and an arc?
Answer:
A chord is a straight line segment joining two points on a circle, while an arc is the curved portion of the circumference between those points. The longest chord of a circle is its diameter. A chord divides a circle into segments, whereas an arc forms part of a sector. Chords are measured in linear units, while arc lengths are measured along the curved boundary. Both concepts are closely related and are used in finding areas and perimeters of sectors and segments in geometry problems.
Q20. How are areas related to circles used in practical situations?
Answer:
Areas related to circles have numerous practical applications. Engineers use them while designing roads, bridges, gears, and circular tracks. Architects apply these concepts in constructing domes, arches, and decorative patterns. Gardeners use circular area calculations for planning parks and fountains. Sports grounds often contain circular regions requiring area measurements. Manufacturers use circle formulas while designing wheels, pipes, and machine components. By understanding sectors, segments, and circular rings, students can solve real-life measurement problems effectively. Hence, areas related to circles are an important topic with wide practical significance.