Circles – Class 10 CBSE Mathematics
Goal – Understand the geometry of circles, remember the main theorems, and solve problems involving areas, arc lengths, tangents, chords, etc.
1. Simple Explanation
A circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the centre).
Key elements of a circle:
| Symbol | Meaning | Formula |
|---|---|---|
| (O) | Centre | – |
| (r) | Radius | – |
| (d) | Diameter | (d = 2r) |
| (AB) | Chord | – |
| (AB) (secant) | Line cutting the circle at two points | – |
| (AT) (tangent) | Line touching the circle at one point | – |
Important facts
- The radius to the point of contact of a tangent is perpendicular to the tangent.
- A chord that is equidistant from the centre is unique (there’s only one such chord).
- The arc is the part of the circumference between two points.
- A sector is the region bounded by two radii and the arc between them.
- A segment is the region bounded by a chord and the arc it subtends.
2. Key Points
2.1 Basic Formulae
| Quantity | Formula | Notes |
|---|---|---|
| Circumference | (C = 2\pi r) | ( \pi \approx 3.1416) |
| Area | (A = \pi r^{2}) | |
| Length of an arc of central angle (\theta) (degrees) | (L = \frac{\theta}{360^\circ},C) | Alternatively (L = \frac{\pi r \theta}{180^\circ}) |
| Area of a sector (angle (\theta^\circ)) | (A_{\text{sector}} = \frac{\theta}{360^\circ},\pi r^{2}) |
2.2 Tangent–Chord Theorem
-
Statement: If a tangent (AT) touches the circle at (T) and cuts a chord (AB) at (A), then
[ \angle (AT, AB) = \angle (BT, AT) ] i.e., the angle between the tangent and a chord equals the angle in the alternate segment. -
Consequence: Tangent is perpendicular to radius: [ \angle OTA = 90^\circ ]
2.3 Intersecting Chords Theorem
If two chords (AB) and (CD) intersect at (P) inside the circle: [ AP \cdot PB = CP \cdot PD ]
2.4 Secant–Tangent Theorem
If a secant (AB) and a tangent (AC) meet at point (A) outside the circle: [ (AB)(AC) = (AD)(AE) ] where (D,E) are the intersection points of the secant with the circle.
2.5 Segment Area
Area of a segment (bounded by a chord (AB) and the arc (AB)): [ A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} ] where (A_{\text{sector}}) uses the central angle subtended by (AB), and (A_{\text{triangle}}) is the area of (\triangle OAB).
2.6 Common Mistakes to Avoid
| Mistake | Why it’s wrong |
|---|---|
| Forgetting that the radius to the point of contact of a tangent is perpendicular | Leads to wrong angle calculations |
| Using degrees where radians are required (or vice‑versa) | Wrong numerical values |
| Confusing chord length with arc length | They are different |
| Applying the intersecting chords theorem only when chords intersect outside the circle | The theorem holds only for inside intersection |
3. Worked Example
Problem
A circle has area (400\pi \text{ cm}^{2}).
Find:
- The radius (r).
- The circumference (C).
- The length of an arc subtended by a central angle of (120^\circ).
Solution
-
Find radius
[ A = \pi r^{2} ;\Rightarrow; 400\pi = \pi r^{2} ] Divide both sides by (\pi): [ 400 = r^{2} ;\Rightarrow; r = \sqrt{400} = 20 \text{ cm} ]
-
Circumference
[ C = 2\pi r = 2\pi(20) = 40\pi \text{ cm} ]
-
**Arc length for (120