🧠 Introduction

In this chapter, we move beyond basic geometry to calculate areas and perimeters of circular shapes. You’ll learn how to find the area of:

  • Circles

  • Sectors (a slice of a circle)

  • Segments (a part of a circle bounded by a chord and arc)

You'll also apply these formulas to solve real-life problems involving paths, wheels, designs, etc.

🔵 1. Perimeter and Area of a Circle

Let the radius of a circle be rr, and diameter be d=2rd = 2r.

  • Circumference (Perimeter) = 2πr2\pi r

  • Area of a circle = πr2\pi r^2

Use π=227\pi = \frac{22}{7} or 3.14 as instructed.

🟣 2. Area of Sector of a Circle

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

If the angle at the center is θ\theta:

  • Area of sector = θ360∘×πr2\frac{\theta}{360^\circ} \times \pi r^2

  • Length of arc = θ360∘×2πr\frac{\theta}{360^\circ} \times 2\pi r

💡 Use this when the circle is divided into portions like slices of pizza or pie.

🔶 3. Area of a Segment of a Circle

A segment is the area enclosed between a chord and the arc.

To find it:

  • Area of segment = Area of sector – Area of triangle

📌 Steps:

  1. Calculate sector area using θ360×πr2\frac{\theta}{360} \times \pi r^2

  2. Calculate triangle area using formulas like:

    • 12absin⁡C\frac{1}{2}ab \sin C (if two sides and included angle are known)

    • Heron’s formula (for general triangle)

🟩 4. Areas of Combinations of Plane Figures

You'll often get complex figures made of circles, semicircles, quadrants, or a mix of shapes.

🧮 Approach:

  • Break into simpler shapes (rectangle, circle, semicircle, triangle)

  • Find individual areas

  • Add or subtract to get the desired region

🧠 Example: Find shaded area in a square with a circle inscribed.

📘 Sample Problems

Q1. Find the area of a sector of radius 7 cm and angle 60°.

  • Area=60360×227×72=25.66 cm2\text{Area} = \frac{60}{360} \times \frac{22}{7} \times 7^2 = 25.66 \text{ cm}^2

Q2. A wheel of radius 28 cm makes 500 revolutions. How much distance does it cover?

  • Circumference = 2πr=2×227×28=176 cm2\pi r = 2 \times \frac{22}{7} \times 28 = 176 \text{ cm}

  • Distance = 176×500=88,000 cm=880 m176 \times 500 = 88,000 \text{ cm} = 880 \text{ m}

Q3. Find the area of a segment of a circle with radius 10 cm and angle 90°.

  • Sector area = 90360×πr2=78.5 cm2\frac{90}{360} \times \pi r^2 = 78.5 \text{ cm}^2

  • Triangle area = 12×10×10=50 cm2\frac{1}{2} \times 10 \times 10 = 50 \text{ cm}^2

  • Segment area = 78.5−50=28.5 cm278.5 - 50 = 28.5 \text{ cm}^2

📝 Key Formulas

QuantityFormula
Circumference2πr2\pi r
Area of circleπr2\pi r^2
Area of sectorθ360×πr2\frac{\theta}{360} \times \pi r^2
Length of arcθ360×2πr\frac{\theta}{360} \times 2\pi r
Area of segmentSector area – Triangle area

📌 Real-Life Applications

  • Designing logos, fans, wheels, paths

  • Calculating distance traveled by tires

  • Estimating land area in circular fields or plots

📥 Download Chapter 12 PDF Notes: Coming Soon
📘 Previous Chapter: Chapter 11 – Constructions »
📚 Next Chapter: Chapter 13 – Surface Areas and Volumes »