🟠 Class 10 Maths Chapter 12: Areas Related to Circles – Notes & Blog Post

 

🧠 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.

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🔵 1. Perimeter and Area of a Circle

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

• Circumference (Perimeter) = $2\pi r$

• Area of a circle = $\pi r^2$

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

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🟣 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 $\theta$:

• Area of sector = $\frac{\theta}{360^\circ} \times \pi r^2$

• Length of arc = $\frac{\theta}{360^\circ} \times 2\pi r$

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

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🔶 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 $\frac{\theta}{360} \times \pi r^2$

2. Calculate triangle area using formulas like:

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

   • Heron’s formula (for general triangle)

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🟩 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.

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📘 Sample Problems

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

• $\text{Area} = \frac{60}{360} \times \frac{22}{7} \times 7^2 = 25.66 \text{ cm}^2$

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

• Circumference = $2\pi r = 2 \times \frac{22}{7} \times 28 = 176 \text{ cm}$

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

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

• Sector area = $\frac{90}{360} \times \pi r^2 = 78.5 \text{ cm}^2$

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

• Segment area = $78.5 - 50 = 28.5 \text{ cm}^2$

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📝 Key Formulas

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📌 Real-Life Applications

• Designing logos, fans, wheels, paths

• Calculating distance traveled by tires

• Estimating land area in circular fields or plots

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