SECTORS • SEGMENTS • SHADED REGIONS
Areas Related to Circles
Worksheet with Online Quiz
Shade the region, trace the boundary and choose the right formula. Build confidence with circle areas, arcs, sectors and segments.
See the Region
Distinguish a sector from a segment before calculating.
Explore shaded diagrams →Choose the Formula
Connect angles, radii, areas and curved lengths.
Try the calculation labs →Apply Your Learning
Explore wheels, clock hands and garden planning.
Take the next challenge →Areas Related to Circles Online Quiz
Question 1 of 20 • Easy • Answered 0/20
A sector is bounded by…
Quiz instructions
Attempt 10 easy, five average and five HOTS questions in 20 minutes. The timer starts with your first answer or question navigation. An answer opens the next question. Use Previous or the numbered buttons to revisit questions. One mark per correct answer with no negative marking. Solutions appear after submission or timeout.
Interactive Circle Area Games
Touch the Region
Tap the requested feature or use the labelled buttons.
Tap the curved arc AB.
Choose your game
Tap a diagram, adjust a sector angle, match formulas, sort units or arrange calculation steps. Use the raised selector to change games. Each set contains five tasks.
Sector Segment and Shaded Region Activities
Answer entry and π
All numerical interactive tasks use π=22/7. Enter numbers without units. You may type 14π, 14*pi, 7/2 or 49*sqrt(3). Here π and pi are evaluated as 22/7 for the stated calculation convention. Give non-terminating decimals to at least three places.
Extra Practice with Circular Areas
Major Sector Explorer
Use the complementary angle 360°−θ. Use π=22/7.
Minor angle=45° and radius=7 cm. Find the major sector area and major arc length.
Use π=22/7. Enter fractions or expressions such as 14π, 14*pi or 49*sqrt(3). For decimals, use at least three decimal places.
Explore the extra sets
Practise major sectors, rolling wheels, clock motion, garden costs and creating valid models. Five tasks are available in each set. Entries stay available while the page remains open.
Revision Tips for Students
- Decide whether the question asks for area, curved length or the complete boundary.
- Convert a diameter to a radius before substitution.
- For sectors, use the fraction θ/360 of the whole circle.
- For a sector perimeter, add both radii to the arc length.
- For a minor segment, subtract the triangle from its sector.
- For a major segment, subtract the minor segment from the whole circle.
- Use the stated value of π consistently and keep radicals until the final step.
- Convert all measurements to the same unit and write area in square units.
Teaching Tips for Parents and Teachers
- Ask students to shade the target region before choosing a formula.
- Trace the entire boundary to distinguish perimeter from arc length.
- Use the slider to compare equal-radius sectors at different angles.
- Compare two examples with equal angles but different radii.
- Ask students to explain why a segment needs a triangle-area subtraction.
Areas Related to Circles Notes Formulas and Solved Questions
This Areas Related to Circles Class 10 worksheet connects diagrams with calculations. Use the quiz to check your understanding of sector area, arc length, segment area and shaded figures. The interactive activities place each question beside its diagram and answer controls. Use the existing worksheet download option on this page for written practice.
Difference Between a Sector and a Segment
A sector is enclosed by two radii and their corresponding arc. A segment is enclosed by a chord and its arc. For a minor central angle, the minor sector contains the centre-to-chord triangle and the minor segment. Removing that triangle leaves the minor segment.
Areas Related to Circles Class 10 Formula Sheet
Let r be radius, θ the central angle in degrees, l arc length and A sector area.
| Measurement | Formula |
|---|---|
| Circle area | πr² |
| Circumference | 2πr |
| Sector area | A=(θ/360)πr² |
| Arc length | l=(θ/360)2πr |
| Sector perimeter for 0°<θ<360° | 2r+l |
| Area from arc length | A=(1/2)rl |
| Minor segment area | Minor sector area−triangle area |
| Major segment area | πr²−minor segment area |
| Annulus area | π(R²−r²), where R>r |
| Semicircle perimeter | πr+2r |
Segment Area Formulas for 60° 90° and 120°
For a minor segment, subtract the triangle from its sector. The triangle area is (√3/4)r² for 60°, r²/2 for 90°, and (√3/4)r² for 120°. The 120° triangle is not equilateral, even though its area has the same expression as the 60° triangle. You can obtain the triangle areas by drawing a perpendicular from the centre to the chord.
Areas Related to Circles Class 10 Important Questions
Practise sector area, arc length, complete sector perimeter, segment area, shaded combinations, major regions and missing measurements. Applications include a rolling wheel, a clock hand sweeping a sector and a sprinkler covering a garden. These question types provide useful revision coverage; they do not predict an upcoming paper.
Areas Related to Circles Class 10 Examples with Solutions
Find sector area and arc length
A sector has radius 21 cm and angle 60°. Use π=22/7.
A=(60/360)×(22/7)×21²
A=(1/6)×1386
A=231 cm²
l=(60/360)×2×(22/7)×21
l=22 cm
Find the complete sector perimeter
A 90° sector has radius 21 cm. Use π=22/7.
l=(90/360)×2×(22/7)×21
l=33 cm
P=2r+l
P=42+33
P=75 cm
Find a minor segment area
A chord subtends 90° at the centre of a circle of radius 42 cm. Use π=22/7.
Sector area=(90/360)×(22/7)×42²
Sector area=1386 cm²
Triangle area=(1/2)×42×42
Triangle area=882 cm²
Segment area=1386−882
Segment area=504 cm²
Areas Related to Circles Class 10 Case-Based Questions
Plan a sprinkler zone
A sprinkler reaches 14 m and rotates through 120°. Treatment costs ₹3 per square metre. Use π=22/7.
What shape is watered? A sector, because the sprinkler sweeps an angle around a fixed centre.
Watered area=(120/360)×(22/7)×14²
Watered area=616/3 m²
Cost=(616/3)×3
Cost=₹616
What is the curved outer boundary?
l=(120/360)×2×(22/7)×14
l=88/3 m
Areas Related to Circles Class 10 Extra Questions with Answers
Use π=22/7 for these calculations.
- Find the 60° arc length for radius 14 cm.
View solution
l=(60/360)×2×(22/7)×14
l=44/3 cm
- Find the major sector area when the minor angle is 60° and radius is 7 cm.
View solution
Major angle=360°−60°
Major angle=300°
A=(300/360)×154
A=385/3 cm²
- Find the circle area for radius 10.5 cm.
View solution
A=(22/7)×10.5²
A=346.5 cm²
- Find the annulus area for outer radius 28 cm and inner radius 21 cm.
View solution
A=(22/7)×(28²−21²)
A=(22/7)×343
A=1078 cm²
- A sector has area 132 cm² and arc length 22 cm. Find radius.
View solution
r=2A/l
r=(2×132)/22
r=12 cm
How to Approach NCERT Exercise 11.1
Identify the requested region and mark the radius and angle. For shaded figures, write the larger area and the part to subtract before calculating. For perimeter, trace only the exposed boundary. Check the exact question against your textbook edition when using exercise solutions. The examples here are original practice, not a complete reproduction of NCERT solutions.
Current Syllabus and Official Sample-Paper Update
The official CBSE Mathematics syllabus for 2026–27 includes sector and segment areas and related perimeter problems. It restricts segment-area calculations to central angles of 60°, 90° and 120°. The current official Mathematics Standard sample paper includes questions relating sector area, arc length and radius. Practise A=(1/2)rl as well as the angle-based formulas. These are verified sample-paper themes rather than a ranking of repeated board-exam questions.
Common Mistakes in Circular Area Questions
Do not use diameter in place of radius, confuse an arc with a sector perimeter, subtract a triangle for a sector question, or forget to subtract it for a segment question. An annulus uses a difference of squared radii, not the square of their difference. Use the problem’s chosen approximation for π consistently.
Related Maths Worksheets and Revision
Areas Related to Circles Frequently Asked Questions
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 its arc. For a minor segment, subtract the triangle between the radii and chord from the minor sector.
How do I find the area of a sector?
Multiply the whole circle area πr² by θ/360, where θ is the central angle in degrees. For a 90° sector, take one quarter of the circle area.
What is the difference between arc length and sector perimeter?
Arc length measures only the curved edge. For a sector with angle strictly between 0° and 360°, its complete perimeter is the arc plus two radii. A whole circle has circumference 2πr.
How do I find a minor segment area?
Calculate the relevant minor sector area, calculate the triangle area between its two radii and chord, and subtract. For 90°, the triangle is right-angled at the centre.
How do I calculate a major segment area?
Subtract the corresponding minor segment from the whole circle. Subtracting only the minor sector would leave a major sector instead.
How can I find sector area when only radius and arc length are given?
Use A=(1/2)rl. For example, a radius of 12 cm and arc length of 11 cm give area 66 cm². Both lengths must be in the same unit.
How do I solve shaded-region questions?
Break the figure into familiar shapes. Label the outer area and the parts removed, then add or subtract them without counting overlaps twice. For perimeter, trace the exposed outline separately.
Which Areas Related to Circles questions are important for exams?
Practise sector area, arc length, sector perimeter, minor segments, major regions, shaded combinations and inverse questions. Use official sample papers for verified examples and practise explaining which boundary or region the question asks for.
Should I use 22/7 or 3.14 for π?
Follow the value stated in the question or paper instructions. If an exact answer in terms of π is requested, leave π in the result. In the numerical activities on this page, use 22/7 as instructed.
How do I find the area of a circular ring?
Subtract the inner circle area from the outer circle area. With outer radius R and inner radius r, area is π(R²−r²), with R greater than r. It is not π(R−r)².
How do I solve clock-hand and rolling-wheel questions?
A minute hand turns 6° per minute. Its tip travels an arc while the hand sweeps a sector. For a wheel rolling without slipping, divide travelled distance by circumference after converting both lengths to the same unit.
What is the perimeter of a semicircle?
Add its curved arc πr to its diameter 2r. The result is πr+2r. Using only half the circumference omits the straight edge.
Can You Identify the Region Before Calculating?
Choose the area, trace the boundary and test your understanding.
Start the Online Quiz
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