CBSE Class 10 Mathematics • Chapter 11 • Revision 2026
Areas Related to Circles Class 10
Revise circular regions with our Areas Related to Circles Class 10 notes and mind map PDF. The PDF contains notes and a mind map for organising your revision. Use the explanations below to understand sector area, arc length, segment area and the difference between area and perimeter before solving questions.
A pizza slice, the path swept by a clock hand and a circular garden all involve parts of circles. In Class 10 Maths Areas Related to Circles, you learn to measure these regions by identifying the radius, central angle and shape involved. The most useful starting point is simple: a sector occupies the same fraction of a circle’s area as its angle occupies of a complete 360° turn.
These revision notes bring together the important formulas, definitions, solved examples and common student doubts. Read them alongside your textbook, then use the mind map to recall the connections between circles, sectors and segments.
Areas Related to Circles Class 10 Notes: Key Terms
- Radius: The distance from the centre to any point on the circle.
- Diameter: A chord passing through the centre; its length is twice the radius.
- Arc: A portion of the circumference between two points.
- Chord: A line segment joining two points on the circle.
- Sector: A region bounded by two radii and their corresponding arc.
- Segment: A region bounded by a chord and its corresponding arc.
- Central angle: The angle formed by two radii at the centre.
Sector and Segment: How to Tell Them Apart
Imagine a slice of pizza reaching the centre: its two straight sides are radii, so it represents a sector. Now imagine cutting across the pizza along a straight chord and keeping the curved piece at the edge: that region represents a segment.
For a minor segment, join the chord’s endpoints to the centre. This creates a sector containing a triangle. Minor segment area = minor sector area − triangle area.
Areas Related to Circles Class 10 All Formulas
In this formula sheet, r is the radius, d is the diameter and θ is the central angle measured in degrees. For a circular ring, R is the outer radius and r is the inner radius.
| Quantity | Formula | What to Remember |
|---|---|---|
| Diameter | d = 2r | Divide the diameter by 2 to find the radius. |
| Circumference | 2πr = πd | Measures the complete curved boundary. |
| Area of a circle | πr2 | Measures the region inside the circle. |
| Arc length | (θ ÷ 360) × 2πr | Includes only the curved part. |
| Area of a sector | (θ ÷ 360) × πr2 | Multiply the full circle’s area by its angle fraction. |
| Perimeter of a sector | 2r + arc length | Includes both radii and the curved boundary. |
| Area of a minor segment | Minor sector area − corresponding triangle area | Find the triangle’s area separately. |
| Area of a major sector | πr2 − minor sector area | Its angle is 360° − θ. |
| Area of a major segment | πr2 − minor segment area | Subtract the smaller segment from the full circle. |
| Area of a semicircle | πr2 ÷ 2 | A semicircle corresponds to 180°. |
| Perimeter of a semicircle | πr + 2r | Add the diameter to the semicircular arc. |
| Area of a quadrant | πr2 ÷ 4 | A quadrant corresponds to 90°. |
| Area of a circular ring | π(R2 − r2) | Subtract the inner circle’s area from the outer circle’s area. |
Why Do We Divide the Angle by 360?
A full circle is 360°. A 90° sector occupies 90 ÷ 360 = 1/4 of the circle, so it has one-quarter of the circle’s area and one-quarter of its circumference as its arc length. A 60° sector similarly represents one-sixth of the circle.
Areas Related to Circles Class 10 Solved Examples
Example 1: Find Sector Area, Arc Length and Perimeter
A sector has radius 21 cm and central angle 120°. Find its area, arc length and perimeter. Use π = 22/7.
Sector area:
Area = (120 ÷ 360) × (22/7) × 212
= (1/3) × (22/7) × 441
= 462 cm2
Arc length:
Arc length = (120 ÷ 360) × 2 × (22/7) × 21
= (1/3) × 132
= 44 cm
Sector perimeter:
Perimeter = 2r + arc length
= 42 + 44
= 86 cm
Example 2: Find the Area of a Minor Segment
A chord subtends an angle of 90° at the centre of a circle of radius 7 cm. Find the minor segment’s area. Use π = 22/7.
Sector area = (90 ÷ 360) × (22/7) × 72
= (1/4) × 154
= 38.5 cm2
The two radii form a right-angled triangle.
Triangle area = (1/2) × 7 × 7
= 24.5 cm2
Minor segment area = sector area − triangle area
= 38.5 − 24.5
= 14 cm2
The corresponding major segment has area:
154 − 14 = 140 cm2.
Example 3: Find the Area Swept by a Clock Hand
A minute hand is 14 cm long. Find the area swept by it in 10 minutes. Use π = 22/7.
The minute hand turns 360° in 60 minutes.
Angle swept in 10 minutes = (10 ÷ 60) × 360°
= 60°
Area swept = (60 ÷ 360) × (22/7) × 142
= (1/6) × 616
= 308/3 cm2
≈ 102.67 cm2
Example 4: Find the Area of a Circular Path
A circular garden has radius 7 m. A path 3 m wide surrounds it on the outside. Find the path’s area. Use π = 22/7.
Inner radius = 7 m
Outer radius = 7 + 3 = 10 m
Path area = π(R2 − r2)
= (22/7) × (100 − 49)
= (22/7) × 51
= 1122/7 m2
≈ 160.29 m2
How to Solve Shaded Area Questions
Before calculating, identify exactly which region the question asks you to measure. A shaded region may be a sector, a segment or the difference between two familiar shapes. Write the relationship in words first, such as “shaded area = semicircle area − triangle area”. This makes the calculation easier to check.
- Mark the centre, radii, chord and given angles on the diagram.
- Convert every measurement to the same unit.
- Break the figure into circles, sectors, triangles or rectangles.
- Calculate each required area separately.
- Add non-overlapping regions or subtract the excluded region.
- Write the final answer in square units.
Area and Perimeter Need Different Thinking
For area, consider the region inside the boundary. For perimeter, trace the boundary itself. A minor segment’s perimeter is its chord length plus its minor arc length; the two radii used in the area calculation are not part of that boundary.
Areas Related to Circles Class 10 Mind Map for Revision
Use the Areas Related to Circles Class 10 mind map in your PDF to organise formulas into connected groups. Begin with the full circle, then link each smaller region to the calculation that produces it.
- Circle: Radius, diameter, circumference and area.
- Sector: Central angle, angle fraction and sector area.
- Arc: The same angle fraction multiplied by circumference.
- Segment: Sector area minus the corresponding triangle.
- Major regions: Full circle area minus the minor region.
- Applications: Clock hands, circular paths and shaded figures.
For your 2026 revision, try recreating this structure without looking at the PDF. Then solve one question from each group. This checks whether you can choose the formula as well as remember it.
Important Questions and Quick Practice
Practise a mix of direct formula questions, MCQs and problems that require interpreting a diagram. Use your textbook exercises and verified previous year questions to apply these concepts in different settings.
- Find the area of a 90° sector of radius 14 cm using π = 22/7. Answer: 154 cm2.
- Find the arc length of a 60° sector of radius 21 cm using π = 22/7. Answer: 22 cm.
- Find the perimeter of a semicircle of radius 7 cm using π = 22/7. Answer: 36 cm.
- A sector has radius 10 cm and arc length 12 cm. Find its area. Answer: 60 cm2, using area = (1/2) × radius × arc length.
- If a circle’s radius doubles, what happens to its area? Answer: Its area becomes four times the original area.
Continue Your Class 10 Maths Practice
Apply these formulas with the Areas Related to Circles Class 10 worksheet with solutions . Attempt each question before checking the solution.
Review tangent properties in our Circles Class 10 notes and mind map , or revise triangle relationships with Triangles Class 10 notes . Explore the CBSE Class 10 Maths notes collection for more chapter revision resources.
Frequently Asked Questions
What does the Areas Related to Circles Class 10 PDF contain?
The PDF contains notes and a mind map. Use the notes to revise concepts, then use the mind map to recall how the formulas connect. The worked examples on this page provide additional practice in selecting and applying those formulas.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. For example, a slice reaching the centre is a sector, while a curved region cut off by a chord is a segment.
What is the formula for the area of a sector?
Sector area = (θ ÷ 360) × πr2, where θ is the central angle in degrees. A 90° sector is one-quarter of the full circle, so its area is πr2 ÷ 4.
Is arc length the same as the perimeter of a sector?
Arc length measures only the curved boundary. Sector perimeter includes the arc and both radii. If the radius is 7 cm and the arc length is 11 cm, the sector’s perimeter is 11 + 7 + 7 = 25 cm.
How do I find the area of a minor segment?
Subtract the corresponding triangle’s area from the minor sector’s area. If the sector area is 38.5 cm2 and the triangle area is 24.5 cm2, the minor segment area is 14 cm2.
How do I find the triangle’s area in a segment question?
Use the central angle to identify the triangle. At 90°, the two radii are perpendicular, giving area r2/2. At 60°, the triangle is equilateral, giving area √3r2/4. For a 120° central angle, draw a perpendicular from the centre to the chord and use the two resulting right triangles to find the base and height.
How are major sector and major segment areas calculated?
Major sector area = circle area − minor sector area. Major segment area = circle area − minor segment area. Make sure you subtract the matching type of region.
Should I use 22/7 or 3.14 for π?
Use the value specified in the question. If no approximation is requested, an exact answer containing π may be appropriate. Avoid changing between 22/7 and 3.14 during the same calculation.
How do I calculate a sector’s area when its perimeter is given?
First find arc length = perimeter − 2r. Then use sector area = (1/2) × r × arc length. For example, if the perimeter is 30 cm and the radius is 7 cm, the arc length is 16 cm and the area is (1/2) × 7 × 16 = 56 cm2.
How do I find the angle swept by a clock hand?
A minute hand moves 6° per minute, so in 15 minutes it sweeps 90°. An hour hand moves 30° per hour, or 0.5° per minute. Identify which hand is given before calculating the sector area.
What mistakes should I avoid in this chapter?
Common mistakes include using the diameter as the radius, forgetting the two radii in sector perimeter, confusing sectors with segments and writing area in cm instead of cm2. Keep calculations exact for as long as possible and round only when required.
How should I revise Areas Related to Circles Class 10?
Learn the definitions first, revise the formula table and mind map, then practise one sector, one segment and one application question. Follow this with worksheet questions and textbook exercises. Check your diagram interpretation, calculations and units when reviewing answers.
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