Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Find the radius of a circle whose area is 616 cm2 (use π = 227). cm

Correct! Area = πr2 = 616, so r2 = 616 × 722 = 196, therefore r = 14 cm.

(2) Find the circumference of a circle whose area is 154 cm2 (use π = 227). cm

Perfect! First find r = 7 cm, then circumference = 2πr = 2 × 227 × 7 = 44 cm.

(3) A circular ring has outer radius 14 cm and inner radius 7 cm. Find its area. cm2

Excellent! Area = πR2−r2 = 227196−49 = 227 × 147 = 462 cm2.

(4) Find the radius of a circle whose circumference is 62.8 cm (use π = 3.14). cm

Great! Circumference = 2πr = 62.8, so r = 62.82×3.14 = 10 cm.

(5) Define sector and segment of a circle. Sector is the region enclosed by two radiitwo chords and the between them while segment is the region enclosed by a and the between the chord’s endpointschord's length.

Correct! A sector is a pie-shaped region bounded by two radii and an arc. A segment is the region between a chord and the arc it cuts off.

Short Answer Questions (2 Marks Each)

Answer each question with detailed calculations

(1) A circular park has radius 28 m. A path 4 m wide surrounds it. Find the area of the path.

Area of path: m2

Excellent! Park area = π × 282 = 2464 m2. Total area = π × 322 = 3217.14 m2. Path area = 3217.14 - 2464 = 753.14 m2 ≈ 704 m2 (using π = 227).

(2) A wheel of diameter 1.4 m makes 500 revolutions. Find the distance traveled by the wheel.

Distance: m

Perfect! Circumference = πd = 227 × 1.4 = 4.4 m. Distance = 500 × 4.4 = 2200 m.

(3) A circular plate has radius 14 cm. A smaller circle of radius 7 cm is drawn inside it. Find the area of the ring formed.

Area of ring: cm2

Correct! Area = πR2−r2 = 227196−49 = 462 cm2.

(4) A circular field has radius 35 m. A wire is to be placed along the boundary. Find the length of wire needed (use π = 227).

Length of wire: m

Great! Length = Circumference = 2πr = 2 × 227 × 35 = 220 m.

(5) A circular flower bed has radius 10 m. Find the cost of leveling it at Rs 5 per m2.

Cost: Rs

Perfect! Area = πr2 = 227 × 100 = 314.29 m2. Cost = 314.29 × 5 = Rs 1571.43 ≈ Rs 1570.

Long Answer Questions (4 Marks Each)

(1) A circular pond has diameter 28 m. A path 3 m wide surrounds it. Find the area of the path (using π = 227).

Area of path: m2

Correct! Pond area = π × 14² = 616 m2. Total area = π × 17² = 907.46 m2. Path area = 907.46 - 616 = 291.46 m2 ≈ 297 m2.

(2) A circular field has diameter 84 m. A farmer wants to fence it with three layers of wire. Find the total length of wire needed (use π = 227).

Total wire length: m

Perfect! Circumference = πd = 227 × 84 = 264 m. Total wire = 3 × 264 = 792 m.

(3) A circular garden has radius 21 m. A flower bed in the shape of a smaller circle of radius 7 m is made in the center. Find the area of the remaining garden.

Remaining area: m2

Excellent! Garden area = π × 212 = 1386 m2. Flower bed area = π × 72 = 154 m2. Remaining = 1386 - 154 = 1232 m2.

(4) A wheel of diameter 1.4 m makes 500 revolutions. Find the distance traveled by the wheel in meters.

Distance: m

Great! Circumference = π × 1.4 = 4.4 m. Distance = 500 × 4.4 = 2200 m.

(5) A circular field has radius 35 m. A path 3 m wide is made around it. Find the area of the path and the total area of the field.

Path area: m2 Total area: m2

Correct! Field area = π × 352 = 3857.14 m2. Total area = π × 382 = 4536 m2. Path area = 4536 - 3857.14 = 678.86 m2.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The radius of a circle whose area is 616 cm2 (π = 227) is:

(a) 14 cm (b) 12 cm (c) 13 cm (d) 15 cm

Correct! Area = πr2 = 616, so r2 = 196, therefore r = 14 cm.

(2) The circumference of a circle with area 154 cm2 (π = 227) is:

(a) 44 cm (b) 42 cm (c) 46 cm (d) 48 cm

Correct! First find r = 7 cm, then circumference = 2πr = 44 cm.

(3) The area of a circular ring with outer radius 14 cm and inner radius 7 cm is:

(a) 462 cm2 (b) 450 cm2 (c) 470 cm2 (d) 480 cm2

Correct! Area = πR2−r2 = 227196−49 = 462 cm2.

(4) The radius of a circle whose circumference is 62.8 cm (π = 3.14) is:

(a) 10 cm (b) 9 cm (c) 11 cm (d) 12 cm

Correct! Radius = Circumference ÷ 2π = 62.8 ÷ (2 × 3.14) = 10 cm.

(5) The area of a circle with radius 14 cm is:

(a) 616 cm2 (b) 600 cm2 (c) 620 cm2 (d) 610 cm2

Correct! Area = πr2 = 227 × 196 = 616 cm2.

(6) The distance traveled by a wheel of diameter 1.4 m making 500 revolutions is:

(a) 2200 m (b) 2198 m (c) 2150 m (d) 2250 m

Correct! Circumference = π × 1.4 = 4.4 m. Distance = 500 × 4.4 = 2200 m.

(7) The area of a circular path 3 m wide around a circular pond of radius 21 m is:

(a) 414 m2 (b) 400 m2 (c) 420 m2 (d) 440 m2

Correct! Pond area = π × 212 = 1386 m2. Total area = π × 242 = 1809.56 m2. Path area ≈ 414 m2.

(8) A circular field has radius 35 m. A wire is to be placed along its boundary. Length of wire needed (π = 22/7) is:

(a) 220 m (b) 210 m (c) 230 m (d) 200 m

Correct! Circumference = 2πr = 2 × 227 × 35 = 220 m.

(9) A circular plate has radius 14 cm. A smaller circle of radius 7 cm is drawn inside it. Area of ring is:

(a) 462 cm2 (b) 440 cm2 (c) 450 cm2 (d) 470 cm2

Correct! Area = πR2−r2 = 227196−49 = 462 cm2.

(10) A circular flower bed has radius 10 m. Cost of leveling at Rs 5 per m2 is:

(a) Rs 1570 (b) Rs 1500 (c) Rs 1550 (d) Rs 1600

Correct! Area = π × 100 = 314.29 m2. Cost = 314.29 × 5 ≈ Rs 1570.

Expert Circle Challenge

Determine whether these statements are True or False:

Expert Circle Mastery Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Expert Circles" worksheet and learned:

(1) Advanced Reverse Calculations: Finding radius from complex area measurements using sophisticated algebraic manipulation

(2) Large-scale Geometric Problems: Working confidently with circles having radii over 30 units and areas over 1000 square units

(3) Multi-layer Applications: Solving problems involving multiple circumferences (triple-layer fencing) and complex layering

(4) Professional Cost Calculations: Integrating geometric calculations with economic applications and real-world pricing

(5) Complex Ring and Path Geometry: Managing intricate area difference calculations with large numbers and precision

(6) Industrial Wheel Problems: Understanding large-scale transportation and machinery applications

(7) Sector and Segment Concepts: Advanced circle terminology for professional geometric applications

(8) Precision in Large Calculations: Maintaining accuracy through complex multi-step problems with large measurements

(9) Advanced Problem Decomposition: Breaking down complex real-world scenarios into manageable calculation steps

(10) Professional-level Applications: Solving problems relevant to engineering, construction, and industrial design

Exceptional achievement! You have mastered expert-level circle geometry and can solve complex professional problems requiring advanced mathematical precision and real-world application skills!