Circles Class 10 Definition Properties Theorems and Solved Examples
The concept of Circles for Class 10 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Circles form the base for many geometric properties and theorems that appear in Class 10 board exams and competitive tests. Understanding circles makes geometry easier and helps students with questions in trigonometry, coordinate geometry, and daily reasoning.
What Is Circles for Class 10?
A circle is a two-dimensional closed shape consisting of all points in a plane that are at an equal distance from a fixed point, called the center. In Class 10 mathematics, circles extend beyond simple shapes to cover concepts such as radius, diameter, chord, tangent, sector, and segment. You’ll find this concept applied in areas such as symmetry, equations of circles, and angle properties within circles.
Key Formula for Circles for Class 10
Here’s the standard formula:
Area of a Circle = \( \pi r^2 \ )
Circumference of a Circle = \( 2\pi r \ )
Equation of a Circle (center at (h, k)): \( (x-h)^2 + (y-k)^2 = r^2 \ )
Parts of a Circle (With Table)
| Part Name | Definition |
|---|---|
| Center | The fixed point from which all points on the circle are equidistant. |
| Radius | A line segment from the center to any point on the circle. |
| Diameter | A chord passing through the center; longest chord (twice the radius). |
| Chord | A line segment joining two points on the circle. |
| Arc | A part of the circumference. |
| Sector | A region bounded by two radii and an arc. |
| Segment | A region bounded by a chord and its corresponding arc. |
| Tangent | A line touching the circle at exactly one point. |
| Secant | A line intersecting the circle at two points. |
Major Properties of Circles (Class 10)
- All radii of a circle are equal.
- The diameter is the longest chord and equals twice the radius.
- Equal chords are equidistant from the center.
- The tangent at any point is perpendicular to the radius at that point.
- Chords equidistant from the center are equal in length.
- Two circles are congruent if their radii are equal.
- If two tangents are drawn from an external point, their lengths are equal.
Step-by-Step Illustration: Example Problem
Let’s solve a sample problem commonly seen in Class 10 exams.
Question: From a point 25 cm away from the center of a circle (radius = 7 cm), find the length of the tangent drawn to the circle.
1. Let O be the center, OP = 25 cm, radius (OA) = 7 cm.2. OA is perpendicular to the tangent at A; consider triangle OAP (right-angled at A).
3. By Pythagoras Theorem:
\( PA^2 = OP^2 - OA^2 \)
4. Substitute values:
\( PA^2 = (25)^2 - (7)^2 = 625 - 49 = 576 \)
5. \( PA = \sqrt{576} = 24 \) cm
Final Answer: The length of the tangent is 24 cm.
Cross-Disciplinary Usage
Circles for Class 10 are not only essential in Maths but also play an important role in Physics (circular motion), Computer Science (graphics, coordinate geometry), and logical reasoning. Concepts like tangent and geometry formulas appear in JEE, NEET, and Olympiad problems. Circles represent wheels, coins, clocks, and many objects in everyday life.
Speed Trick or Vedic Shortcut
When quickly calculating circumference, use π ≈ 22/7 for easy division, especially with radii that are multiples of 7. For example, if r = 14 cm:
2 × 22/7 × 14 = 2 × 22 × 2 = 88 cm (circumference – no calculator needed!).
Common Errors and Misunderstandings
- Mixing up circle and disc: the circle is the boundary, the disc is the area inside.
- Confusing chord and diameter — diameter always passes through the center.
- Miscalculating area by forgetting to square the radius.
- Assuming tangent passes through the circle instead of touching at one point.
Interlinks For More Learning (Vedantu)
- Parts of Circle – See detailed diagrams and learn each part for exam-labeling.
- Area of Circle – Formulas, stepwise explanations, and solved practice problems.
- Circumference of a Circle – Steps to calculate perimeter with quick tricks.
- Circle Theorems – List and proofs of key theorems for higher-order problems.
Relation to Other Concepts
Understanding circles for Class 10 unlocks the door to mastering areas related to circles, coordinate geometry, and advanced theorems in mathematics. Mastering the properties of tangents and chords also makes trigonometry much easier in higher classes.
Quick Classroom Tip
To remember all major parts: “CRaD TASeS” (Center, Radius, Diameter, Tangent, Arc, Segment, Sector, Secant). Draw and label a circle each time you revise.
We explored Circles for Class 10—from definition, parts, formula, properties, solved examples, and important connections to other chapters. For in-depth theory, solved problems, and expert-prepared revision, keep practicing with Vedantu and ace your exams with confidence!
FAQs on Circles Class 10 Maths Complete Guide with Theorems and Problems
1. What is a circle in Class 10 Maths?
A circle is the set of all points in a plane that are at a fixed distance from a fixed point called the centre. The fixed distance is known as the radius.
- The fixed point is called the centre.
- The fixed distance is called the radius.
- A circle is a simple closed curve.
2. What is the formula for the circumference of a circle?
The formula for the circumference of a circle is C = 2πr, where r is the radius. It can also be written as C = πd, where d is the diameter.
- π (pi) ≈ 3.14 or 22/7
- r = radius
- d = 2r
3. What is the formula for the area of a circle?
The formula for the area of a circle is A = πr², where r is the radius. This formula is widely used in mensuration problems.
- π ≈ 3.14 or 22/7
- Square the radius before multiplying by π
4. What is the difference between a chord and a diameter?
A chord is any line segment joining two points on a circle, while a diameter is the longest chord that passes through the centre.
- Every diameter is a chord.
- Not every chord is a diameter.
- Diameter = 2 × radius.
5. What is the theorem of tangent to a circle in Class 10?
The tangent to a circle theorem states that the tangent at any point of a circle is perpendicular to the radius at the point of contact. In other words, the angle between the radius and the tangent is 90°.
- Radius ⟂ Tangent at point of contact
- Used to prove many geometry results
6. How many tangents can be drawn from a point to a circle?
From an external point, exactly two tangents can be drawn to a circle. The lengths of these tangents from the same external point are equal.
- If the point is outside → 2 tangents
- If the point is on the circle → 1 tangent
- If the point is inside → No tangent
7. Why are tangents from an external point equal?
Tangents drawn from an external point to a circle are equal in length because they form congruent right triangles with the radii. If PA and PB are tangents from point P, then PA = PB.
- Radius is perpendicular to tangent.
- Common hypotenuse is formed.
- Triangles are congruent by RHS.
8. What is a tangent to a circle?
A tangent to a circle is a line that touches the circle at exactly one point. That point is called the point of contact.
- It intersects the circle at only one point.
- It is perpendicular to the radius at that point.
9. What is the relationship between radius and diameter?
The diameter of a circle is twice the radius, that is d = 2r. Conversely, the radius is half of the diameter: r = d/2.
- Diameter passes through the centre.
- It is the longest chord of the circle.
10. How do you prove that a tangent is perpendicular to the radius?
To prove that a tangent is perpendicular to the radius, show that the angle between them is 90° at the point of contact. The proof is based on the fact that the shortest distance from the centre to a line is the perpendicular.
- Join centre O to point of contact P.
- Assume the line is not perpendicular.
- Then a shorter distance can be drawn, which is impossible.
