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Area of Circle and Sector

The area of a circle is the surface inside its boundary, and the area of a sector is a fractional part of the circle's area.

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Student-friendly explanation

Circle area depends on the square of radius. A sector's area depends on the central angle, just as arc length depends on the same fraction of the full circle.

How to write this in exams

  1. 1

    Start with the exact idea

    The area of a circle is the surface inside its boundary, and the area of a sector is a fractional part of the circle's area.

  2. 2

    Then show how to use it

    Identify radius, square it, multiply by π, and for a sector multiply by θ/360.

  3. 3

    Add one concrete example

    For radius 7 cm, circle area = 22/7 × 7 × 7 = 154 cm². A 90° sector has area 90/360 × 154 = 38.5 cm².

  4. 4

    Avoid this incomplete answer

    44 cm² for radius 7 cm is wrong because 44 cm is circumference, not area.

Definition

The area of a circle is the surface inside its boundary, and the area of a sector is a fractional part of the circle's area.

Example

For radius 7 cm, circle area = 22/7 × 7 × 7 = 154 cm². A 90° sector has area 90/360 × 154 = 38.5 cm².

Rule to remember

Circle area = πr². Sector area = θ/360 × πr², where θ is the central angle in degrees.

Memory hook

Area fills the circle, so radius is multiplied twice.

Examples and method

Worked example

A sector has radius 14 cm and angle 45°. Area = 45/360 × 22/7 × 14 × 14 = 77 cm².

Method to apply

Identify radius, square it, multiply by π, and for a sector multiply by θ/360.

Diagram support

A circle or sector diagram should mark radius and central angle clearly.

How CBSE asks it

Questions may ask area of a circular park, sector-shaped sheet, shaded region, or a comparison with arc length.

Avoid common mistakes

Common confusion

Students use circumference formula when the question asks for the surface inside a circle.

Common wrong answer

44 cm² for radius 7 cm is wrong because 44 cm is circumference, not area.

Exam tip

Area uses square units and r²; circumference or arc length uses length units.

Quick check

What is the area of a circle of radius 10 cm using π = 3.14?

The area is πr² = 3.14 × 10 × 10 = 314 cm² because area depends on the square of radius.

Answer writing and exam use

1-mark answer

The area of a circle is the surface inside its boundary, and the area of a sector is a fractional part of the circle's area.

2-mark answer

The area of a circle is the surface inside its boundary, and the area of a sector is a fractional part of the circle's area. Circle area = πr². Sector area = θ/360 × πr², where θ is the central angle in degrees. For radius 7 cm, circle area = 22/7 × 7 × 7 = 154 cm². A 90° sector has area 90/360 × 154 = 38.5 cm².

3-mark answer

Circle area depends on the square of radius. A sector's area depends on the central angle, just as arc length depends on the same fraction of the full circle. Circle area = πr². Sector area = θ/360 × πr², where θ is the central angle in degrees. A sector has radius 14 cm and angle 45°. Area = 45/360 × 22/7 × 14 × 14 = 77 cm². Questions may ask area of a circular park, sector-shaped sheet, shaded region, or a comparison with arc length. 44 cm² for radius 7 cm is wrong because 44 cm is circumference, not area.
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