Important Properties and Theorems of Circles in Geometry
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 Circle with Formulas, Diagrams & Examples
1. What are the main properties of a circle in maths?
The main properties of a circle center around its consistent distance from a central point. Key properties include:
- All points on the circumference are equidistant from the center.
- The radius is the distance from the center to any point on the circumference.
- The diameter is a chord passing through the center; it's twice the length of the radius.
- A chord is a line segment connecting any two points on the circumference.
- A tangent is a line that touches the circle at exactly one point.
- A secant is a line intersecting the circle at two points.
- The circle possesses infinite lines of symmetry, each passing through the center.
2. How is the radius of a circle defined?
The radius of a circle is defined as the distance from the center of the circle to any point on its circumference. All radii of a given circle are equal in length.
3. What is the formula for the circumference of a circle?
The formula for the circumference (C) of a circle is given by C = 2πr, where 'r' represents the radius of the circle and π (pi) is a mathematical constant approximately equal to 3.14159.
4. What is special about tangents to a circle?
A tangent to a circle is a line that intersects the circle at exactly one point, called the point of tangency. The radius drawn to the point of tangency is always perpendicular to the tangent.
5. How many axes of symmetry does a circle have?
A circle has an infinite number of axes of symmetry. Each axis of symmetry is a diameter of the circle, passing through the center.
6. What is the relationship between a circle's diameter and its radius?
The diameter of a circle is always twice the length of its radius. Mathematically, Diameter = 2 × Radius.
7. What is a chord in a circle?
A chord is a straight line segment whose endpoints both lie on the circle's circumference. The longest possible chord is the diameter.
8. How is the area of a circle calculated?
The area (A) of a circle is calculated using the formula A = πr², where 'r' represents the radius of the circle.
9. What is a secant of a circle?
A secant is a line that intersects a circle at two distinct points. It extends beyond the circle, unlike a chord which is contained entirely within the circle.
10. What is an arc of a circle?
An arc is a portion of the circumference of a circle. It's a curved line segment connecting two points on the circle.
11. What are some real-world examples of circles?
Circles are found everywhere! Examples include: wheels, coins, the sun, clock faces, pizzas, and many more objects with a perfectly round shape.
12. Explain the concept of concyclic points.
Concyclic points are points that lie on the same circle. If four or more points are concyclic, a circle can be drawn that passes through all of them.
