What are the main parts of a circle?
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 Circle Definition for Class 10 – Concept, Diagram, Properties & Formulas
1. What is the formal definition of a circle as per the Class 10 curriculum?
A circle is defined as the collection of all points in a plane that are at a fixed distance from a fixed point in the same plane. The fixed point is called the centre of the circle, and the fixed distance is called the radius. This fundamental definition is the basis for understanding all properties and theorems related to circles in Class 10 Maths for the 2025-26 session.
2. What is the practical difference between a tangent and a secant of a circle?
The key difference lies in how they intersect the circle. A secant is a line that intersects a circle at two distinct points. In contrast, a tangent is a special line that touches the circle at exactly one point. This single point is known as the point of contact. While a secant passes through the interior of the circle, a tangent only grazes its boundary without entering it.
3. What are the two primary theorems related to tangents in the Class 10 Circles chapter?
The two main theorems for the CBSE Class 10 syllabus are:
Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
These theorems are crucial for solving most problems involving tangents.
4. How many tangents can be drawn to a circle from a single point?
The number of tangents depends on the position of the point relative to the circle:
If the point is inside the circle, no tangents can be drawn.
If the point is on the circle, exactly one tangent can be drawn through that point.
If the point is outside the circle, exactly two tangents can be drawn to the circle from that point.
5. What are the most important formulas related to a circle for Class 10?
While the Class 10 chapter on Circles focuses on geometric properties, students must know these fundamental formulas:
Area of a circle: A = πr², where 'r' is the radius.
Circumference (Perimeter) of a circle: C = 2πr, where 'r' is the radius.
Length of an arc of a sector: (θ/360°) × 2πr, where 'θ' is the angle of the sector.
Area of a sector of a circle: (θ/360°) × πr², where 'θ' is the angle of the sector.
6. Why is a tangent considered a special case of a secant?
Understanding a tangent as a special case of a secant helps in visualising its unique properties. Imagine a secant that intersects a circle at two points. If you move this secant such that the two intersection points gradually come closer and closer together, they will eventually coincide to become a single point. The line at this exact position is the tangent. This concept clarifies why a tangent touches the circle at only one point and is fundamental to proving circle theorems.
7. How can the theorem about equal tangents from an external point be applied in geometry?
This theorem has powerful applications. For instance, if a quadrilateral is drawn to circumscribe a circle (meaning all its sides touch the circle), you can prove that the sum of its opposite sides is equal (e.g., AB + CD = AD + BC). This is solved by applying the theorem to each vertex of the quadrilateral, which acts as an external point from which two equal tangents (the sides) are drawn.
8. What are some real-world examples that illustrate the concept of a tangent to a circle?
Tangents appear in many real-world mechanical and natural systems. A common example is a bicycle chain running over a circular gear; the straight parts of the chain are tangent to the gear. Another example is a ball rolling on a flat surface; the point where the ball touches the ground at any instant is a point of tangency.
9. Why is the radius always perpendicular to the tangent at the point of contact?
The radius at the point of contact represents the shortest distance from the centre of the circle to the tangent line. In geometry, the shortest distance from a point to a line is always the perpendicular distance. Any other point on the tangent line would be farther from the centre than the radius, proving that the radius must be perpendicular to the tangent at that specific point of contact.
10. What is a common misconception about the relationship between a chord and a diameter?
A common misconception is failing to recognise that a diameter is a type of chord, not a separate entity. A chord is any line segment whose endpoints both lie on the circle. The diameter is simply the longest possible chord that must pass through the centre of the circle. Therefore, while every diameter is a chord, not every chord is a diameter.
