Assertion (A): A wire is bent into a semicircular shape with radius 14 cm. Using π = 22/7, the length of the wire is 44 cm. Reason (R): The length of a semicircular arc equals πr.

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A rectangle has length 20 m and width 10 m. (a) Calculate its perimeter. (b) If a square has the same perimeter as this rectangle, find the length of its side. (c) If the rectangle’s boundary is reshaped into a circle, what will be its radius? (Take π = 3.14) Show all steps.

(a) 60 m. (b) 15 m. (c) Approximately 9.55 m.

In a circular running track, the inner lane (Lane 1) has a radius of 36 m at the curved portion. Each lane is 1.25 m wide. For a race that covers exactly one full semicircular arc (half of the circle) on the curved part, compute the stagger for Lane 2, i.e., how far ahead the runner in Lane 2 should start compared to Lane 1 so that both cover the same distance on the curved part. Assume the runners stay in the middle of their lanes. Give your reasoning and the stagger length to the nearest centimetre. Take π = 3.14.

The stagger is approximately 3.93 m (or 393 cm).

A standard 400 m running track consists of two straight sections each 100 m long and two semicircular bends. The inner edge of the track is exactly 400 m. The track has 8 lanes, each of width 1.22 m. In a 400 m race, runners in outer lanes start ahead of those in inner lanes to ensure they all run the same distance. This head start is called stagger. (1) Calculate the inner radius of the semicircular bends so that the inner edge totals 400 m. (Use π = 3.14) (2) Explain why the stagger required for Lane 2 relative to Lane 1 is equal to 2π × lane width. (3) Calculate the stagger for Lane 2 when the lane width is 1.22 m. (Use π = 3.14)

Inner radius ≈ 31.85 m; stagger for Lane 2 = 7.66 m; explanation: extra distance is 2π × lane width.

Class 9 Maths Chapter 6: Measuring Space: Perimeter and Area — Important Questions & Sample Paper

Q: What is the constant π, and why is it important in this chapter?

π (pi) is the ratio of any circle's circumference to its diameter, approximately 3.14 or 22/7. It is used to derive formulas for circumference (2πr) and area (πr²) of circles, and to solve problems involving circular tracks and paths.

Q: How do you calculate the stagger for runners in different lanes on a circular track?

Stagger is the extra distance for outer lanes on the curved part. For one full circle, stagger = 2π × (lane width) per lane. For a semi-circle, stagger = π × (lane width). The straight lengths are equal for all lanes.

Q: What is the difference between finding perimeter and area of composite figures that include circles?

Perimeter is the total boundary length, summing straight edges and arc lengths (e.g., semicircle = πr, quarter-circle = (πr)/2). Area is the space enclosed, computed by adding or subtracting areas of simple shapes. For paths, often use subtraction: area(outer) – area(inner).

Q: What common mistakes should students avoid when solving problems from this chapter?

Mistakes include confusing radius and diameter, using wrong value of π (use 22/7 or 3.14 as given), forgetting to halve/double dimensions for semi-circles, and unit errors (ensure consistency in cm/m). Always check if the problem asks for perimeter or area.

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Yes — this page has 44+ original Class 9 Mathematics Chapter 6 (“Measuring Space: Perimeter and Area”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 6 of Class 9 Ganita Manjari, 'Measuring Space: Perimeter and Area,' deepens students' understanding of circles and composite figures. The chapter begins with the constant π, the ratio of circumference to diameter, and then focuses on calculating the perimeter (circumference) and area of circles, semi-circles, and quarter-circles. Students learn the formulas circumference = 2πr and area = πr², and they apply these to real-world contexts such as circular parks, running tracks, and pathways. Key skills include finding the area of circular rings when a path is constructed inside or outside a garden, determining the length of paths that combine straight edges and curved arcs, and calculating stagger distances in multi-lane tracks to ensure fair race distances on curved segments. Typical exam questions ask learners to compute circumferences, areas, and path areas, often using π as 22/7 or 3.14. Multi-step problems may involve parts of circles, such as semi-circular arcs or quarter-circles at corners of rectangles. Understanding these concepts builds a foundation for higher-level geometry and mensuration, and the chapter’s applications make it both practical and engaging for students.

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Mathematics — Measuring Space: Perimeter and Area

Class 9Time: 3 hrsMax Marks: 80

SECTION A

1.
A wire is bent to form a circle of radius 14 cm. If the same wire is straightened and bent into a square, what is the side of the square? (use π = 22/7)
(a) 11 cm(b) 22 cm(c) 33 cm(d) 44 cm
1
2.
Archimedes used inscribed and circumscribed hexagons to show that 3 < π < 2√3. From this, which of the following must be true?
(a) π > 22/7(b) π < 3.14(c) 3 < π < 3.464(d) π is exactly 3.14
1
3.
In a 400 m running track, the lanes are 1 m wide. The difference in the length of one full circle of the outer lane compared to the inner lane is approximately (use π ≈ 3.14):
(a) 3.14 m(b) 6.28 m(c) 1 m(d) 12.56 m
1

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question type	Marks each	In our bank
Multiple Choice (MCQ)	1 mark	13
Assertion–Reason	1 mark	6
Short Answer	2 marks	8
Short Answer	3 marks	6
Long Answer	5 marks	5
Case Study	4 marks	6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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Q1. A wire is bent to form a circle of radius 14 cm. If the same wire is straightened and bent into a square, what is the side of the square? (use π = 22/7)
1 mark
Multiple Choice (MCQ)
(A) 11 cm(B) 22 cm(C) 33 cm(D) 44 cm
▸ Answer▾ Answer
22 cm
Q2. Archimedes used inscribed and circumscribed hexagons to show that 3 < π < 2√3. From this, which of the following must be true?
1 mark
Multiple Choice (MCQ)
(A) π > 22/7(B) π < 3.14(C) 3 < π < 3.464(D) π is exactly 3.14
▸ Answer▾ Answer
3 < π < 3.464
Q3. In a 400 m running track, the lanes are 1 m wide. The difference in the length of one full circle of the outer lane compared to the inner lane is approximately (use π ≈ 3.14):
1 mark
Multiple Choice (MCQ)
(A) 3.14 m(B) 6.28 m(C) 1 m(D) 12.56 m
▸ Answer▾ Answer
6.28 m
Q4. In a 200 m track designed with two straight segments of 50 m each and two semicircular ends, the inner radius of the semicircles is (use π = 22/7):
1 mark
Multiple Choice (MCQ)
(A) 175/11 m(B) 350/11 m(C) 100/7 m(D) 200/7 m
▸ Answer▾ Answer
175/11 m
Q5. Assertion (A): A wire is bent into a semicircular shape with radius 14 cm. Using π = 22/7, the length of the wire is 44 cm. Reason (R): The length of a semicircular arc equals πr.
1 mark
Assertion–Reason
(A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
▸ Answer▾ Answer
Both A and R are true and R is the correct explanation of A.
Q6. Find the circumference of a circle whose radius is 5 cm. (Use π = 3.14)
2 marks
Short Answer
▸ Answer▾ Answer
31.4 cm
Q7. A circular running track has an inner lane of radius 30 m. Each additional lane is 1 m wide. How much longer is one full lap in the second lane than in the inner lane? (Use π = 3.14)
2 marks
Short Answer
▸ Answer▾ Answer
6.28 m
Q8. The length of a semicircular arc is 22 cm. Find the radius of the circle. (Use π = 22/7)
3 marks
Short Answer
▸ Answer▾ Answer
7 cm
Q9. In a 400 m running track, the inner boundary of lane 1 consists of two straight segments of length 84.39 m each and two semicircles with inner radius 36.5 m. Each lane is 1.22 m wide. To ensure all runners cover the same distance in one lap, by how many metres should the start line for lane 2 be ahead of lane 1? (Use π = 3.14 and round to two decimal places.)
3 marks
Short Answer
▸ Answer▾ Answer
7.66 m
Q10. A rectangle has length 20 m and width 10 m. (a) Calculate its perimeter. (b) If a square has the same perimeter as this rectangle, find the length of its side. (c) If the rectangle’s boundary is reshaped into a circle, what will be its radius? (Take π = 3.14) Show all steps.
5 marks
Long Answer
▸ Answer▾ Answer
(a) 60 m. (b) 15 m. (c) Approximately 9.55 m.
Q11. In a circular running track, the inner lane (Lane 1) has a radius of 36 m at the curved portion. Each lane is 1.25 m wide. For a race that covers exactly one full semicircular arc (half of the circle) on the curved part, compute the stagger for Lane 2, i.e., how far ahead the runner in Lane 2 should start compared to Lane 1 so that both cover the same distance on the curved part. Assume the runners stay in the middle of their lanes. Give your reasoning and the stagger length to the nearest centimetre. Take π = 3.14.
5 marks
Long Answer
▸ Answer▾ Answer
The stagger is approximately 3.93 m (or 393 cm).
Q12. A standard 400 m running track consists of two straight sections each 100 m long and two semicircular bends. The inner edge of the track is exactly 400 m. The track has 8 lanes, each of width 1.22 m. In a 400 m race, runners in outer lanes start ahead of those in inner lanes to ensure they all run the same distance. This head start is called stagger.
4 marks
Case Study
1. (i) Calculate the inner radius of the semicircular bends so that the inner edge totals 400 m. (Use π = 3.14)1 mark
2. (ii) Explain why the stagger required for Lane 2 relative to Lane 1 is equal to 2π × lane width.1 mark
3. (iii) Calculate the stagger for Lane 2 when the lane width is 1.22 m. (Use π = 3.14)2 marks
▸ Answer▾ Answer
Inner radius ≈ 31.85 m; stagger for Lane 2 = 7.66 m; explanation: extra distance is 2π × lane width.

Frequently asked questions

What is the constant π, and why is it important in this chapter?

π (pi) is the ratio of any circle's circumference to its diameter, approximately 3.14 or 22/7. It is used to derive formulas for circumference (2πr) and area (πr²) of circles, and to solve problems involving circular tracks and paths.

How do you calculate the stagger for runners in different lanes on a circular track?