Areas Related to Circles Class 10 Worksheet with Solutions
Practise circular areas, sectors, segments and shaded figures with this Areas Related to Circles Class 10 worksheet. Use the formula sheet, solved examples and extra questions below to strengthen your understanding before checking the worksheet solutions.
A circular garden, a slice of pizza and the region swept by a clock hand all involve the same mathematical idea: finding the area of a whole circle or a fraction of it. In Areas Related to Circles Class 10, you learn to connect the radius and central angle with arc length, sector area and segment area. These skills also help you calculate shaded regions formed by combining circles, triangles and squares.
When working through a worksheet on areas related to circles for Class 10, begin by identifying the required region. Decide whether the question asks for an area, an arc length or a complete perimeter. Then write the appropriate formula and substitute the values. This simple habit makes both routine calculations and application-based questions easier to solve.
Areas Related to Circles Class 10 Notes and Key Concepts
- Radius: The distance from the centre of a circle to its circumference.
- Diameter: A chord through the centre, equal to twice the radius.
- Arc: A portion of the circumference.
- Sector: The region enclosed by two radii and their connecting arc.
- Segment: The region enclosed by a chord and its corresponding arc.
- Quadrant: A quarter of a circle, with a central angle of 90°.
- Semicircle: Half a circle, corresponding to a central angle of 180°.
The distinction between a sector and a segment is especially important. A sector reaches the centre of the circle, whereas a segment is bounded by a chord. For a minor segment, subtract the area of the triangle formed by the two radii and the chord from the area of the corresponding minor sector.
Areas Related to Circles Class 10 Formulas
Use this Areas Related to Circles Class 10 formula sheet during revision. Here, r is the radius, θ is the central angle measured in degrees, and R is the outer radius of a circular ring.
| Quantity | Formula | Remember |
|---|---|---|
| Area of a circle | πr2 | Express the answer in square units. |
| Circumference | 2πr | This is the complete circular boundary. |
| Arc length | (θ ÷ 360) × 2πr | Use the angle corresponding to the required arc. |
| Area of a sector | (θ ÷ 360) × πr2 | A sector is a fraction of the whole circle. |
| Perimeter of a sector | 2r + arc length | Include both straight radii. |
| Area of a minor segment | Minor sector area − central triangle area | Find the triangle area from the given geometry. |
| Area of a semicircle | πr2 ÷ 2 | Half the area of a circle. |
| Perimeter of a semicircle | πr + 2r | Include the diameter as well as the curved boundary. |
| Area of a circular ring | π(R2 − r2) | Subtract the inner circle from the outer circle. |
Choose the Value of π Carefully
Use the value specified in the question, such as 22/7 or 3.14. If no approximation is requested, an exact answer containing π may be suitable. Keep the same value throughout a calculation and convert all measurements to matching units before substituting.
Areas Related to Circles Class 10 Questions with Solutions
Example 1: Sector Area, Arc Length and Perimeter
Question: A sector has radius 14 cm and central angle 90°. Find its area, arc length and perimeter. Take π = 22/7.
Solution:
Sector area = (90 ÷ 360) × (22 ÷ 7) × 142
Sector area = 154 cm2
Arc length = (90 ÷ 360) × 2 × (22 ÷ 7) × 14
Arc length = 22 cm
Sector perimeter = 2 × 14 + 22
Sector perimeter = 50 cm
Example 2: Area of a Minor Segment
Question: A chord subtends an angle of 90° at the centre of a circle of radius 14 cm. Find the area of the minor segment. Take π = 22/7.
Solution:
Area of the 90° sector = 154 cm2
The two radii form a right-angled triangle.
Triangle area = ½ × 14 × 14 = 98 cm2
Minor segment area = 154 − 98
Minor segment area = 56 cm2
This example shows why the formulas for a sector and a segment cannot be used interchangeably.
Example 3: A Sector with a 120° Angle
Question: Find the area of a sector of radius 21 cm and central angle 120°. Take π = 22/7.
Solution:
Sector area = (120 ÷ 360) × (22 ÷ 7) × 212
Sector area = ⅓ × 1386
Sector area = 462 cm2
To find the corresponding minor segment, you would also need to calculate and subtract the area of the triangle between the two radii.
Case-Based Questions and Real-Life Applications
Areas Related to Circles Class 10 case study questions often describe a clock, garden, umbrella, brooch or windscreen wiper. Translate the situation into a diagram before calculating. The length of a clock hand becomes the radius; equally spaced umbrella ribs divide the circle into equal sectors; a wiper may sweep a sector or an annular sector, depending on the blade’s position.
Clock Case Study: Area Swept by a Minute Hand
Question: A minute hand is 7 cm long. Find the area it sweeps in 15 minutes. Take π = 22/7.
Solution:
The minute hand rotates 360° in 60 minutes.
Angle swept in 15 minutes = (15 ÷ 60) × 360° = 90°
Area swept = (90 ÷ 360) × (22 ÷ 7) × 72
Area swept = 38.5 cm2
Extension: In 30 minutes, the hand sweeps a semicircle, giving an area of 77 cm2.
Horse Grazing at the Corner of a Field
Question: A horse is tied inside a rectangular field at a corner with a 7 m rope. Both adjacent sides are longer than 7 m, and the grazing region has no obstacles. Find the accessible area. Take π = 22/7.
Solution:
The horse can graze within a quadrant of radius 7 m.
Grazing area = ¼ × (22 ÷ 7) × 72
Grazing area = 38.5 m2
Areas Related to Circles Class 10 MCQ with Answers
-
A 90° sector occupies what fraction of a circle?
A. ½ B. ¼ C. ⅓ D. ¾
Answer: B. The fraction is 90/360 = ¼. -
Which expression gives the perimeter of a semicircle?
A. πr B. 2πr C. πr + 2r D. πr2/2
Answer: C. Add the curved boundary and the diameter. -
What angle does a minute hand sweep in 10 minutes?
A. 30° B. 60° C. 90° D. 120°
Answer: B. It moves 6° per minute, so 10 × 6° = 60°. -
If a circle’s radius doubles, its area becomes:
A. Twice B. Three times C. Four times D. Unchanged
Answer: C. π(2r)2 = 4πr2.
Extra Practice Questions and HOTS
Try these Areas Related to Circles Class 10 extra questions with answers after revising the formulas. Use π = 22/7 wherever a numerical approximation is needed.
-
Find the area of a circle with diameter 14 cm.
Answer: Radius = 7 cm; area = 154 cm2. -
Find the arc length of a 60° sector with radius 21 cm.
Answer: 22 cm. -
Find the area of a ring with outer radius 14 cm and inner radius 7 cm.
Answer: 462 cm2. -
A circle of radius 7 cm is inscribed in a square of side 14 cm.
Find the area inside the square but outside the circle.
Answer: 196 − 154 = 42 cm2. -
A sector has radius 10 cm and arc length 8 cm. Find its area
without first calculating the central angle.
Answer: Sector area = ½ × radius × arc length = ½ × 10 × 8 = 40 cm2.
How to Approach Shaded-Region Questions
Identify the complete outer shape, then subtract the unshaded parts. For overlapping figures, check whether any area has been counted twice. Mark radii, diameters and angles on the diagram. Finish by checking that your answer is positive and smaller than the area of the complete outer figure.
NCERT Exercise Practice and Chapter Revision
Use this page alongside your Areas Related to Circles Class 10 NCERT exercises. Exercise numbering can differ between textbook editions: students may search for Exercise 11.1 or older references such as Exercises 12.1, 12.2 and 12.3. Match the question’s wording and diagram with your own textbook rather than relying only on the exercise number.
For a useful revision sequence, study the formulas, solve the worked examples independently, attempt the worksheet and review your mistakes. Then practise NCERT Exemplar problems, competency-based questions and verified previous year questions. Pay particular attention to the difference between length units and square units.
Related Notes, Mind Maps and Worksheets
- Areas Related to Circles Class 10 Notes and Mind Map for concept revision and a chapter overview.
- Circles Class 10 Notes and Mind Map to revise related circle geometry.
- Circles Class 10 Worksheet with Solutions and MCQs for tangent-based practice.
- Triangles Class 10 Worksheet with Solutions to strengthen the triangle calculations used in segment problems.
Frequently Asked Questions
What is the difference between a sector and a segment?
A sector is enclosed by two radii and an arc. A segment is enclosed by a chord and an arc. For example, a quarter-circle is a sector; removing its central right-angled triangle leaves the corresponding minor segment.
What are the most important Areas Related to Circles Class 10 formulas?
Revise circle area πr2, circumference 2πr, sector area (θ/360) × πr2 and arc length (θ/360) × 2πr. Also remember that a sector’s perimeter includes two radii, and a minor segment’s area is the minor sector area minus its central triangle area.
How do I calculate the area of a minor segment?
Calculate the corresponding minor sector area, then subtract the triangle formed by the two radii and the chord. With radius 14 cm and angle 90°, the sector area is 154 cm2 and the triangle area is 98 cm2, giving 56 cm2 when π = 22/7.
How do I find the area of a major sector or major segment?
Subtract the minor sector area from the whole circle’s area to find the major sector. Similarly, subtract the minor segment area from the whole circle’s area to find the major segment. For a minor central angle θ, the corresponding major sector angle is 360° − θ.
Why is arc length different from the perimeter of a sector?
Arc length measures only the curved edge. The perimeter of a sector includes that arc and the two straight radii. For a 90° sector of radius 14 cm, the arc is 22 cm but the complete perimeter is 22 + 14 + 14 = 50 cm, using π = 22/7.
How do I solve minute-hand and hour-hand questions?
Convert the elapsed time into an angle, then use the sector-area formula. A minute hand moves 6° per minute; an hour hand moves 30° per hour, or 0.5° per minute. The hand’s length is the radius. For a single sweep of up to one revolution, use the corresponding angle directly.
How are umbrella and wiper questions solved?
For an umbrella with equally spaced ribs, divide 360° by the number of equal spaces to find each sector angle. For a wiper sweeping from the pivot to radius r, calculate a sector area. If the blade covers only the region between inner radius r and outer radius R, use (θ/360) × π(R2 − r2).
How can I practise Areas Related to Circles Class 10 HOTS questions?
Practise questions involving unknown angles, shaded composite figures, changing radii and the relationship between sector area and arc length. Explain your choice of formula before calculating. For example, doubling a radius multiplies area by four, while doubling the central angle at a fixed radius doubles sector area.
Should I use 22/7 or 3.14 for π?
Follow the value stated in the question. If the answer is requested in terms of π, leave π in the final expression. Avoid changing approximations midway through a solution, because this can produce inconsistent answers.
What is the weightage of Areas Related to Circles in Class 10?
Do not assume a fixed chapter-wise mark allocation for every paper. Check your board’s applicable syllabus, sample paper and marking scheme. Prepare sector, segment, arc-length and application questions so you can handle different question formats.
Where can I revise Areas Related to Circles Class 10 notes and a mind map?
Visit the Areas Related to Circles Class 10 notes and mind map page, then return to this worksheet for practice. Attempt each question before checking its solution to identify the steps you need to revise.
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