NOTES

📘 Circles Class 10 Notes (Chapter 10)

🔷 Introduction

Circles are an important part of geometry. In simple words, a circle is a round shape. Every point on the circle stays at the same distance from one fixed point. Therefore, a circle has a perfect balance in shape.

Moreover, we see circles in daily life. For example, wheels, clocks, coins, rings, and plates all have circular shapes. As a result, this chapter becomes easy to connect with real situations.

---

🔷 What is a Circle?

A circle is a group of points that stay equally far from one fixed point. We call this fixed point the center.

For instance, if point O is the center, then every point on the circle stays at the same distance from O. Thus, the center controls the whole shape.

---

🔷 Important Parts of a Circle

Understanding the parts of a circle makes this chapter easier. Therefore, learn each term carefully.

---

✔ Center

The fixed point inside the circle is called the center.

Example:
Point O

Moreover, every measurement starts from the center.

---

✔ Radius

A line joining the center to any point on the circle is called the radius.

If O is the center and A lies on the circle:$$OA=r$$OA=r

Therefore, all radii in one circle are equal.

---

✔ Diameter

A line joining two points of the circle through the center is called the diameter.

Formula:$$d=2r$$d=2r

In other words, the diameter is twice the radius.

---

✔ Chord

A line joining any two points on the circle is called a chord.

In addition, the diameter is also a chord. Most importantly, it is the longest chord.

---

✔ Arc

A curved part of the circle is called an arc.

For example, a small curved boundary between two points forms an arc.

---

✔ Sector

Two radii and one arc form a sector.

Similarly, a pizza slice looks like a sector.

---

✔ Segment

A chord and an arc together form a segment.

Thus, segments create separate regions inside the circle.

---

🔷 Tangent to a Circle

A tangent touches the circle at exactly one point. It never cuts the circle.

We call that touching point the point of contact.

Therefore, a tangent has only one common point with the circle.

---

🔷 Important Theorems

These theorems help students solve many questions. So, remember them carefully.

---

Theorem 1

A tangent always makes a right angle with the radius at the point of contact.

If OA is radius and AB is tangent:$$OA\perp AB$$OA⊥AB

Therefore:$$\angle OAB=90^\circ$$∠OAB=90∘

Hence, radius and tangent always meet at 90°.

---

Theorem 2

Two tangents from one external point always have equal lengths.

If PA and PB are tangents from point P:$$PA=PB$$PA=PB

As a result, both tangent lengths stay equal.

---

🔷 Solved Example

Example

A circle has radius 7 cm. Find its diameter.

Solution

Given:

Radius = 7 cm

Now use:$$d=2r$$d=2r $$d=2\times7$$d=2×7 $$d=14cm$$d=14cm

Answer:

Diameter = 14 cm

---

🔷 Important Properties

Understanding these properties makes problem-solving easier.

• All radii are equal.
• The diameter is twice the radius.
• The diameter is the longest chord.
• A tangent touches the circle at one point.
• Radius and tangent form a right angle.
• Two tangents from one point are equal.

Therefore, these rules appear in many exam questions.

---

🔷 Real-Life Uses

We use circles in many places. For example, circles appear in:

• Wheels
• Clocks
• Coins
• Rings
• Plates
• Gears
• Sports tracks

Moreover, engineers, designers, and builders use circular shapes every day. Thus, circles have many practical uses.