Important Properties of Circle with Formulas and Solved Examples
The concept of properties of circle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding the unique circle properties helps students solve geometry questions faster and with more confidence.
What Is Properties of Circle?
A circle is defined as the set of all points in a plane that are at a fixed distance (called the radius) from a fixed point (called the centre). You’ll find this concept applied in areas such as circle geometry, circle theorems, and real-life measurement.
Key Properties of Circle
Here are the standard properties of a circle every student should know:
- All points on the circle are equidistant from the centre.
- The diameter is always twice the radius.
- The longest chord in a circle is its diameter.
- The perpendicular bisector of any chord passes through the centre.
- Equal chords are equidistant from the centre.
- All tangents to a circle are perpendicular to the radius at the point of contact.
- A circle has infinite lines of symmetry (it is perfectly symmetrical).
| Property | Meaning | Example/Note |
|---|---|---|
| Radius | Distance from centre to any point on circle | OA, OB etc. |
| Diameter | Passes through centre, longest chord | AB if O is midpoint |
| Chord | Segment joining any 2 points | PQ, RS etc. |
| Tangent | Touches the circle at only one point | Will always be perpendicular to the radius |
| Symmetry | Circle looks the same from any angle | Infinite axes of symmetry |
Key Formula for Properties of Circle
Here’s the standard list of formulas related to properties of circle:
| Name | Formula | Terms |
|---|---|---|
| Circumference | C = 2πr | r = radius |
| Area | A = πr2 | r = radius |
| Diameter | D = 2r | r = radius |
| Arc Length | L = (θ/360)×2πr | θ = angle in degrees |
Important Circle Theorems
- The angle subtended by a diameter at the circle’s circumference is always 90°.
- Equal chords subtend equal angles at the centre.
- Tangents drawn from an external point to a circle are equal in length.
- If a radius is drawn to the point of contact of a tangent, they are perpendicular.
- If two chords are equal, they are equidistant from the centre.
Step-by-Step Illustration
- Given: Find area and circumference of a circle with diameter 10 cm.
Radius = Diameter ÷ 2 = 10 ÷ 2 = 5 cm
- Area = π r2 = 3.14 × 5 × 5 = 78.5 cm2
- Circumference = 2 π r = 2 × 3.14 × 5 = 31.4 cm
Frequent Errors and Misunderstandings
- Confusing diameter and radius (remember, diameter is always twice the radius).
- Forgetting that the diameter is the longest chord.
- Assuming a tangent can cut through the circle (it only touches at one point).
- Thinking that all chords pass through the centre (only the diameter does).
Relation to Other Concepts
The idea of properties of circle connects closely with chord properties, tangent properties, and circle theorems. Mastering these basics makes topics like equation of a circle and area of circle much easier in higher grades.
Classroom Tip
A quick way to remember circle properties: draw and label the circle’s centre, radius, diameter, chord, and tangent on paper. This helps visualize each property clearly. Vedantu’s teachers use interactive diagrams to make these concepts super easy in their live classes.
Try These Yourself
- What is the diameter if the radius of a circle is 7 cm?
- If the chord of a circle is 8 cm from the centre, what can you say about all other chords at this distance?
- Does a tangent have any part inside the circle?
- Find the circumference if the diameter = 14 cm (π = 22/7).
Wrapping It All Up
We explored properties of circle—from basic definition and formula to the most important theorems, examples, and errors. Practice these regularly, and use resources from Vedantu for clear diagrams, solved questions, and live classes. Mastering circle properties sets you up for success in all geometry topics!
Explore More on Circles
FAQs on Properties of a Circle Explained with Diagrams and Proofs
1. What are the main properties of a circle?
The main properties of a circle describe the relationships between its radius, diameter, chords, arcs, tangents, and angles. A circle is the set of all points in a plane at a fixed distance from a fixed point called the center.
- All radii of a circle are equal.
- The diameter is twice the radius: d = 2r.
- The longest chord of a circle is the diameter.
- A tangent to a circle is perpendicular to the radius at the point of contact.
- Angles subtended by the same arc at the circumference are equal.
2. What is the formula for the circumference of a circle?
The circumference of a circle is given by the formula C = 2πr or C = πd. Here, r is the radius and d is the diameter.
- If r = 7 cm, then C = 2π × 7 = 14π cm.
- Using π ≈ 22/7, C = 14 × 22/7 = 44 cm.
3. What is the formula for the area of a circle?
The area of a circle is calculated using the formula A = πr². Here, r is the radius of the circle.
- If r = 5 cm, then A = π × 5² = 25π.
- Using π ≈ 3.14, A = 25 × 3.14 = 78.5 cm².
4. What is the relationship between radius and diameter in a circle?
The diameter of a circle is always twice the radius, so the relationship is d = 2r. Conversely, r = d/2.
- If r = 4 cm, then d = 2 × 4 = 8 cm.
- If d = 10 cm, then r = 10/2 = 5 cm.
5. What is a chord in a circle and what are its properties?
A chord of a circle is a line segment joining any two points on the circle. The diameter is the longest chord of the circle.
- Equal chords are equidistant from the center.
- The perpendicular from the center to a chord bisects the chord.
- Chords equidistant from the center are equal in length.
6. What are the properties of a tangent to a circle?
A tangent to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at that point.
- The radius drawn to the point of contact is perpendicular to the tangent.
- Tangents drawn from an external point to a circle are equal in length.
- A circle can have infinitely many tangents.
7. What is the angle subtended by a diameter in a circle?
The angle subtended by a diameter at the circumference of a circle is always 90°. This is known as the angle in a semicircle theorem.
- If AB is a diameter and C is any point on the circle, then ∠ACB = 90°.
8. What are the properties of angles in a circle?
The angle properties of a circle describe relationships between central and inscribed angles formed by arcs and chords.
- The angle at the center is twice the angle at the circumference standing on the same arc.
- Angles in the same segment of a circle are equal.
- Opposite angles of a cyclic quadrilateral sum to 180°.
9. What is a sector and what are its properties?
A sector of a circle is the region enclosed by two radii and the arc between them.
- Area of a sector = (θ/360) × πr², where θ is the central angle in degrees.
- Arc length = (θ/360) × 2πr.
10. What is a cyclic quadrilateral and what is its property?
A cyclic quadrilateral is a four-sided figure whose vertices lie on a circle, and its key property is that opposite angles add up to 180°.
- If ∠A and ∠C are opposite angles, then ∠A + ∠C = 180°.
- If one pair of opposite angles is supplementary, the quadrilateral is cyclic.
