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MathematicsClass 9Measuring Space

Measuring Space | Class 9 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 4 min read

Measuring Space | Class 9 Mathematics Notes

Measuring Space – this guide gives you a concise, exam-ready overview of Measuring Space from Class 9 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

6.4 LENGTH OF AN ARC OF A CIRCLE

This section explains how to calculate the length of an arc of a circle. The circumference of a full circle with radius r is 2πr. A semicircle, being half the circle, has an arc length of πr. Similarly, a quarter circle has an arc length of (πr)/2. These results are derived using symmetry and rotation arguments. The general formula for the length of an arc subtending an angle θ degrees at the center of the circle is given by (2πr × θ/360). This formula is essential in practical applications such as designing athletics tracks, where the curved parts are arcs of circles. The section includes a detailed example of a 400 m athletics track with two straight sections and two semicircular curves, explaining how the total distance run is calculated and how stagger distances compensate for lane differences.

📊 Diagram: Fig. 6.8: Two semicircles making a full circle; Fig. 6.9: Four quarter circles make a full circle; Fig. 6.10: Length of an arc of a circle; Fig. 6.11: Schematic diagram of a 400m athletics track

🧪 Activity: Think and Reflect: Calculate stagger distances between lanes using arc length differences.

🔗 Connection: Leads to problem-solving on perimeters involving arcs and circles.

Frequently asked questions

Two circles of equal radius are located such that each circle passes through the centre of the other circle (Fig. 6.12). Given that the radius of each circle is r units, find the perimeter of the shape formed by the two circles in terms of r units. (Ignore the dotted portions that lie within the circles.)

Let the two circles have centers A and B, each with radius r units. Each circle passes through the center of the other, so AB = r. The circles intersect at points C and D. Triangle ABC is equilateral with sides r, so angle CAB = 60°. The arcs forming the perimeter are each 120° arcs (since the angle subtended by the chord CD at the center is 120°). Each red arc is therefore 1/3 of the circumference of a circle with radius r.

The perimeter consists of two such arcs, so total perimeter = 2 × (1/3

In Fig. 6.13, we see points P and Q and two paths connecting them. The first path is made up of the semicircle a. The other path is made up of three semicircles (b, c and d). Which path is longer? Choose one: (i) Path a is longer. (ii) Path b + c + d is longer. (iii) The two paths have equal length. (Try to answer this before reading on.)

The two paths have equal length.

Explanation: Let the radii of semicircles a, b, c, d be a', b', c', d' respectively. Length of semicircle a = πa'. Length of semicircles b, c, d are πb', πc', πd' respectively. Total length of second path = π(b' + c' + d'). Since the length of PQ = 2a' = 2b' + 2c' + 2d', it follows that a' = b' + c' + d'. Hence, lengths of both paths are equal.

1. The perimeter of a circle is 44 cm. What is its radius?

Given perimeter (circumference) C = 44 cm. We know circumference C = 2πr. Using π = 22/7, 44 = 2 × (22/7) × r => 44 = (44/7) × r => r = 44 × (7/44) = 7 cm. So, the radius of the circle is 7 cm.

2. Calculate, correct to 3 significant figures, the circumference of a circle with: (i) radius 7 cm (ii) radius 10 cm (iii) radius 12 cm.

Formula: Circumference C = 2πr. (i) r = 7 cm C = 2 × (22/7) × 7 = 44 cm (ii) r = 10 cm C = 2 × (22/7) × 10 = (44/7) × 10 = 440/7 ≈ 62.857 cm Rounded to 3 significant figures: 62.9 cm (iii) r = 12 cm C = 2 × (22/7) × 12 = (44/7) × 12 = 528/7 ≈ 75.429 cm Rounded to 3 significant figures: 75.4 cm

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