Areas Related to Circles

1) Hook — A Fun Real-Life Example

Imagine you are helping your grandmother prepare a traditional modak (a sweet dumpling) for Ganesh Chaturthi. The modak is shaped like a cone with a circular base. To wrap the modak perfectly with a circular piece of cloth, you need to know the area of the cloth required. This is exactly where the concept of areas related to circles becomes useful in everyday life!

2) Core Concepts — Explanation with Examples

In Class 10, you learn how to find areas of various parts related to circles:

• Area of a Circle
• Area of a Sector (a “pizza slice” of a circle)
• Area of a Segment (part of a circle cut off by a chord)
• Area of a Ring (the region between two concentric circles)

1. Area of a Circle: If the radius of the circle is r, then its area is

Example: A circular garden has a radius of 7 m. Find its area.

Area = π × 7² = 49π ≈ 153.94 m²

2. Area of a Sector: A sector is a portion of a circle enclosed by two radii and the arc between them.

If the angle of the sector at the centre is θ°, then

Example: Find the area of a sector with radius 14 cm and angle 60°.

Area = (60/360) × π × 14² = (1/6) × π × 196 = (196/6)π ≈ 102.67 cm²

3. Area of a Segment: The segment is the region bounded by a chord and the corresponding arc.

To find the area of a segment:

• Calculate the area of the sector with angle θ°
• Calculate the area of the triangle formed by the two radii and the chord
• Segment area = Sector area − Triangle area

Example: For a circle with radius 10 cm and sector angle 60°, find the area of the segment.

Sector area = (60/360) × π × 10² = (1/6) × 100π ≈ 52.36 cm²

Triangle area = (1/2) × r² × sin θ = (1/2) × 10² × sin 60° = 50 × (√3/2) ≈ 43.30 cm²

Segment area = 52.36 − 43.30 = 9.06 cm²

4. Area of a Ring: A ring is the region between two concentric circles with radii R and r (R > r).

Example: Find the area of a ring formed between two circles of radii 14 cm and 10 cm.

Area = π(14² − 10²) = π(196 − 100) = 96π ≈ 301.59 cm²

3) Key Formulas / Rules

4) Did You Know?

π (Pi) is an irrational number approximately equal to 3.14159, but it has been calculated to over 31 trillion digits beyond the decimal! Ancient Indian mathematician Madhava of Sangamagrama (14th century) was one of the first to discover infinite series to calculate π accurately.

5) Exam Tips — Common Mistakes & Board Patterns

• Always write units (cm², m²) in your final answer.
• Use π ≈ 22/7 or 3.14 as per question instruction; do not mix both.
• For segment area, remember to subtract the triangle area from sector area.
• Check if the angle θ is in degrees before using formulas.
• Common question types: Finding area of sectors, segments, rings, and shaded regions.
• Often, questions combine multiple shapes — draw a neat diagram and mark all dimensions.
• Remember the mnemonic: "Circle’s Area is Pi r Square, Sector’s Part is Theta Share."