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Class 10 · Maths · Chapter 11 · NCERT Class 10 Mathematics

Areas Related to Circles Class 10 Notes

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Chapter mind map

The whole chapter at a glance: the big idea, then each branch and what sits under it.

Areas Related to Circles

Calculating areas and lengths of circular regions, sectors, and segments using radius and central angle θ.

  1. Fundamental Properties

    Basic definitions and constants for circular calculations.

    • Circumference and Area — Circumference is 2πr (linear units); Area is πr² (square units).
    • The Constant Pi (π) — Approximately 22/7 or 3.14; used to relate radius to boundary and area.
    • Unit Conventions — Area is in square units (cm², m²); lengths like radius are in linear units (cm, m).

  2. Sectors and Arc Length

    Regions enclosed by two radii and an arc, resembling a pizza slice.

    • Sector Area Formula — Area = (θ / 360) * πr², where θ is the central angle in degrees.
    • Arc Length (l) — The curved boundary length: l = (θ / 360) * 2πr.
    • Minor vs Major Sectors — Minor is the default; Major is (360 - θ) or Total Area - Minor Sector.

  3. Segments of a Circle

    The region between a chord and its corresponding arc.

    • Minor Segment Area — Calculated as: Area of corresponding Sector - Area of Triangle OAB.
    • Major Segment Area — Calculated as: Total Circle Area - Minor Segment Area.
    • Trigonometric Triangle Area — Use sin(θ/2) for half-chord and cos(θ/2) for height; Area = 1/2 * base * height.
    • Special Case: 120° Angle — Triangle area = r²√3/4. Height is r/2 and chord is r√3.

  4. Practical Applications

    Applying circular geometry to real-world objects and designs.

    • Clock Mechanics — Minute hand sweeps 6° per minute; 5 minutes = 30° sector.
    • Wipers and Lighthouses — Sweeping blades or beams form sectors; area depends on range (radius) and angle.
    • Composite Designs — Calculate area of one repeating segment/sector and multiply by total count.

Chapter notes

A comprehensive guide to calculating the areas of sectors and segments of a circle, including arc lengths and practical applications using the unitary method.

01 · Explore

Fundamentals of Circular Regions

Before exploring specific parts of a circle, it is essential to recall the basic properties of the circular region.

A circle is a collection of all points in a plane that are at a fixed distance from a fixed point called the centre. The distance around the boundary of the circle is known as its circumference, given by the formula 2πr, where r is the radius. The region enclosed by this boundary is the area of the circle, calculated as πr².

In this chapter, we focus on parts of this circular region. We use the constant π, which is approximately 22/7 or 3.14. Unless specified otherwise, we use 22/7 for calculations. It is important to note that area is always measured in square units (such as cm² or m²), while lengths like radius or circumference are measured in linear units (such as cm or m).

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Question

If the circumference of a circle is 22 cm, what is its radius?

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NCERT reference: chapter PDF pages 1, 2.

02 · Explore

Defining Sectors and Segments

A circle can be divided into specific regions based on radii and chords.

A sector is the portion of a circular region enclosed by two radii and the corresponding arc. Imagine a slice of pizza; the two straight edges are the radii, and the crust is the arc. The angle formed between the two radii at the centre is called the angle of the sector, often denoted by θ.

A segment is the portion of the circular region enclosed between a chord and its corresponding arc. Unlike a sector, a segment does not necessarily include the centre of the circle. Both sectors and segments come in two sizes: 'minor' (the smaller part) and 'major' (the larger part).

By convention, when we use the terms 'sector' or 'segment' without a prefix, we are referring to the minor sector or minor segment respectively.

OABSectorABOSegment
A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. For this minor sector, subtract triangle OAB to get the minor segment. Learning sketch; use the labels and stated dimensions, not measurements from the picture.

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What is the angle of a major sector if the minor sector angle is 60°?

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NCERT reference: chapter PDF page 1.

03 · Explore

Calculating the Area of a Sector

The area of a sector is proportional to the angle it subtends at the centre.

To find the area of a sector with radius r and angle θ, we use the unitary method. A full circle can be viewed as a sector with an angle of 360°, having an area of πr². Therefore, a sector with an angle of 1° has an area of (πr²) / 360.

For a sector with an angle θ, the area is (θ / 360) * πr². This formula allows us to calculate the area of any 'slice' of the circle if we know the radius and the central angle.

Finding Sector Area

Area = (θ / 360) * πr²

For a circle with radius 4 cm and sector angle 30°, Area = (30 / 360) * 3.14 * 4 * 4. This simplifies to (1 / 12) * 3.14 * 16 = (4 / 3) * 3.14 = 4.186... cm², which is approximately 4.19 cm².

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Find the area of a sector of a circle with radius 6 cm if the angle of the sector is 60°.

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NCERT reference: chapter PDF pages 2, 3.

04 · Explore

Length of an Arc

Just as the sector area is a fraction of the total area, the arc length is a fraction of the total circumference.

The arc is the curved boundary of a sector. The length of an arc (l) of a sector with angle θ and radius r is determined by the ratio of the sector angle to the total 360° angle of the circle.

Since the total circumference is 2πr, the length of an arc is given by the formula: l = (θ / 360) * 2πr. This represents the linear distance along the curve between the two radii.

Arc Length Calculation

l = (θ / 360) * 2πr

In a circle of radius 21 cm, if an arc subtends 60° at the centre: l = (60 / 360) * 2 * (22/7) * 21 = (1/6) * 2 * 22 * 3 = (1/6) * 132 = 22 cm.

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If the radius is 7 cm and the arc length is 11 cm, find the angle θ.

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NCERT reference: chapter PDF pages 2, 7.

The rest of this chapter

Keep reading Areas Related to Circles, free

  1. Locked: 1. Area of a Segment
  2. Locked: 2. Worked Example: Segment with 120° Angle
  3. Locked: 3. Major Sectors and Major Segments
  4. Locked: 4. Practical Applications and Composite Shapes

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