Circles Class 10 Formulas

This circles class 10 formulas sheet collects the results you need from NCERT Chapter 10 Circles: the tangent to a circle, the number of tangents from a point, the tangent length, and the angle relations between two tangents.

Every formula is grouped by topic, with a symbol table, when-to-use guidance, and three worked examples using fresh numbers. The proofs behind the theorems are on the Circles Class 10 notes page — this sheet quotes the results rather than re-deriving them.

Circles Class 10 Formulas at a Glance

All Formulas, Grouped by Topic

The groups follow the NCERT chapter: Section 10.2 (tangent to a circle) and Section 10.3 (number of tangents from a point), then the angle and circumscribed-figure results used in the exercises.

Tangent to a Circle (Section 10.2)

A tangent to a circle is a line that meets the circle in only one point; that point is the point of contact. A line meeting the circle at two points is a secant, and a line with no common point is non-intersecting (NCERT, p. 145). Fig. 10.1 shows the three positions.

The tangent is the limiting case of the secant: as the two intersection points of a line slide together, the secant becomes a tangent. At any point of a circle there is exactly one tangent, and no more than two tangents can be drawn parallel to a given secant (NCERT, pp. 146–147).

Theorem 10.1. The tangent at any point of a circle is perpendicular to the radius through the point of contact (NCERT, p. 147):

\[ OP \perp XY \]

So at the point of contact, the angle between the radius and the tangent is \( 90^\circ \).

Why it is true (Fig. 10.5): every other point Q of the tangent lies outside the circle, so \( OQ \gt OP \). Since OP is the shortest distance from O to the line XY, it is the perpendicular to XY.

You see the same property in the wheel of Fig. 10.4: the ground acts as a tangent, and the radius through the point of contact stands at right angles to it.

Number of Tangents from a Point (Section 10.3)

Where the point lies decides how many tangents pass through it (NCERT, p. 148), as Fig. 10.6 shows:

• point inside the circle — no tangent;
• point on the circle — one tangent;
• point outside the circle — exactly two tangents.

Theorem 10.2. The lengths of tangents drawn from an external point to a circle are equal (NCERT, p. 149):

\[ PQ = PR \]

Why it holds: the radii \( OQ \) and \( OR \) are equal, \( OP \) is common, and the angles at Q and R are right angles by Theorem 10.1, so the right triangles are congruent by RHS (NCERT, p. 149).

A one-line alternative uses Pythagoras: \( PQ^2 = OP^2 – OQ^2 = OP^2 – OR^2 = PR^2 \), which again gives \( PQ = PR \).

Tangent length from Pythagoras. In the same right triangle, the length of a tangent from an external point is (NCERT, p. 149):

\[ PQ = \sqrt{OP^2 – OQ^2} \]

Here \( OQ \) is the radius to the point of contact, so \( OP \gt OQ \) whenever the point is outside the circle.

Angle bisector property. The centre lies on the bisector of the angle between the two tangents (NCERT, p. 149):

\[ \angle OPQ = \angle OPR \]

Angle Relations Between Two Tangents

In the standard figure with external point T and tangents TP, TQ touching at P and Q (Fig. 10.9), triangle TPQ is isosceles because \( TP = TQ \). The chapter gives two angle results (NCERT, p. 150):

\[ \angle PTQ = 2\angle OPQ \]

\[ \angle TPQ = \angle TQP = 90^\circ – \frac{1}{2}\angle PTQ \]

and, from Exercise 10.2 Q10 (NCERT, p. 152), the angle between the two tangents is supplementary to the angle subtended at the centre by the chord joining the points of contact:

\[ \angle PTQ + \angle POQ = 180^\circ \]

Tangents in Circumscribed Figures

Concentric circles (Example 1, NCERT, p. 150). When a chord AB of the larger circle touches the smaller inner circle at P, the radius OP is perpendicular to the chord, and the perpendicular from the centre bisects the chord:

\[ OP \perp AB \quad \text{and} \quad AP = BP \]

Quadrilateral circumscribing a circle (Exercise 10.2 Q8, NCERT, p. 152). For a quadrilateral ABCD whose four sides all touch one circle:

\[ AB + CD = AD + BC \]

The equality of tangents (Theorem 10.2) works at every vertex of a circumscribed figure: the two tangent segments drawn from any one vertex are equal. Fig. 10.14 shows a triangle circumscribing a circle, the figure used in Exercise 10.2 Q12.

What Each Symbol Means

NCERT reuses the letters P, Q, R in different figures. Before substituting, check which point is the centre, which lies on the circle, and which lies outside it.

When to Use Each Formula

Need another chapter? The full Class 10 Maths formulas list is one click away.

Worked Examples

Example 1: Tangent length from an external point

Example 2: Finding the distance of the external point

Example 3: Angle between two tangents

For step-by-step working of every textbook question, see the NCERT solutions for Class 10 Maths Chapter 10 Circles.

Common Mistakes to Avoid

Frequently Asked Questions

How many tangents can a circle have?

Infinitely many. A tangent can be drawn at every point of the circle, and at any one point there is one and only one tangent (NCERT, p. 147). This is the answer to Exercise 10.1 Q1.

Why is a tangent perpendicular to the radius at the point of contact?

If a point Q on the tangent other than the point of contact P were inside the circle, the line would meet the circle twice and be a secant. So every point Q on the tangent gives \( OQ \gt OP \), making P the closest point of the line to the centre.

The shortest distance from a point to a line is along the perpendicular, so \( OP \perp XY \) (Theorem 10.1, NCERT, p. 147).

Are the two tangents from an external point always equal in length?

Yes. The radii to the points of contact are equal, the hypotenuse OP is common, and both triangles are right-angled by Theorem 10.1, so the triangles are congruent by RHS. Hence \( PQ = PR \) (Theorem 10.2, NCERT, p. 149).

If the chord of contact subtends 100° at the centre, what is the angle between the tangents?

80°. The two angles are supplementary, so the angle between the tangents is \( 180^\circ – 100^\circ = 80^\circ \).

Every subject’s formula sheets are organised in the maths formulas index.

Reference: NCERT Class 10 Mathematics textbook (Rationalised NCERT), chapter 10 Circles. Verify the chapter in the official textbook at ncert.nic.in.