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Circles Notes

Home / Class Notes / Class 10 / Math / Circles
✓ Free notes Class 10 Math 2 study files

About these notes

Class 10 Math Circles CBSE notes give concise theorem statements, chord and arc formulas, diagram tips and links to chapter test and MCQ practice on Merit Spring ROH. Main sub-topics: definitions (radius, diameter, chord, arc, sector), angle in the same segment, central vs. inscribed angles, tangent properties and tangent–chord theorem, perpendicular from centre to chord, chord-length formula, cyclic quadrilaterals and related proofs. Suitable for Class 10 students (CBSE / Bihar Board) preparing for board exams, chapter tests, revision and mock tests.

Key points you will learn

  • Class 10 Math Circles CBSE: radius, diameter, chord, arc, sector — know precise definitions
  • Tangent ⟂ radius at point of contact (90°) and tangent–chord theorem: ∠ between tangent and chord = angle in opposite arc
  • Perpendicular from centre to a chord bisects the chord and its arcs (use to form right triangles)
  • Chord length formula: chord = 2√(r² − d²) where d is distance from centre to chord
  • Equal chords subtend equal central angles; converse also true
  • Opposite angles of a cyclic quadrilateral are supplementary (sum 180°)
  • Problem tip: draw radii, mark equal lengths, reduce to right triangles and apply Pythagoras
  • If a parallelogram is circumscribed about a circle, all four sides are equal (it is a rhombus)

What is inside this chapter

📄

Circles

📥 Open notes
📄

Circles

📥 Open notes

Questions students ask

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

8 cm. The perpendicular from the common centre to the chord is 3 cm, so half the chord = √(5² − 3²) = 4 cm; full chord = 2×4 = 8 cm. Draw the right triangle with hypotenuse 5 cm (radius) to check.

What is the angle between a tangent and the radius at the point of contact?

90°. The tangent is perpendicular to the radius at the point of contact, so the angle between them is a right angle; use this to form right triangles in tangent problems.

How do I use 'perpendicular from centre to chord' in problems?

Use it to create right triangles: the perpendicular bisects the chord so half-chord = √(r² − d²) with r the radius and d the distance from centre; this finds chord lengths or subtended angles.

Prove that a parallelogram circumscribing a circle is a rhombus.

Because tangents from a vertex to the circle are equal, each vertex gives equal adjacent tangent segments; in a parallelogram opposite sides equal plus equal tangent lengths force all four sides equal, so it is a rhombus.

Where can I practise MCQ and chapter test for Circles notes?

Merit Spring ROH provides chapter test and MCQ practice linked from the Circles notes page; use those for timed revision, then retry problems by drawing accurate diagrams and applying the standard theorems.

← Previous chapter Some Applications of Trigonometry

More Math chapters

Real Numbers Polynomials Pair of Linear Equations in Two Variables Quadratic Equations Arithmetic Progressions Triangles All chapters →

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