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 fundamental properties that define a circle in geometry?
A circle is a two-dimensional shape defined by a set of core properties. The most important ones are:
- Equidistance: All points on the boundary (circumference) of a circle are at an equal distance from a fixed central point, known as the center.
- Radius (r): The distance from the center to any point on the circumference. All radii in a circle are of equal length.
- Diameter (d): A straight line segment that passes through the center and has its endpoints on the circumference. Its length is always twice the radius (d = 2r).
- Chord: A line segment connecting any two points on the circumference. The diameter is the longest possible chord.
- Symmetry: A circle has infinite lines of symmetry, with each line being a diameter.
2. What are the key formulas associated with a circle and what do they represent?
The two primary formulas for a circle measure its boundary and the space it encloses:
- Circumference (C): This represents the total length of the boundary of the circle. The formula is C = 2πr, where 'r' is the radius and 'π' (pi) is a mathematical constant approximately equal to 3.14159.
- Area (A): This represents the total two-dimensional space inside the circle. The formula is A = πr², where 'r' is the radius. This shows that the area is proportional to the square of its radius.
3. What is the difference between a chord, a secant, and a tangent of a circle?
These three terms describe lines interacting with a circle, but they differ in how they do so:
- A chord is a line segment whose endpoints both lie on the circle's circumference. It is contained entirely within or on the circle.
- A secant is a line that intersects the circle at two distinct points. It is an infinite line that extends outside the circle.
- A tangent is a line that touches the circle at exactly one point, known as the point of tangency. It does not enter the circle's interior.
4. What are some real-world examples that illustrate the properties of a circle?
The properties of circles are visible in many everyday objects and natural phenomena. For example, a bicycle wheel uses the concept of a fixed center (axle) and equal radii (spokes) to ensure smooth rotation. A pizza being cut into equal slices demonstrates sectors. The ripples on a pond after a stone is thrown expand as concentric circles, all sharing the same center.
5. What defines an arc, and how are major and minor arcs distinguished?
An arc is a continuous portion of the circumference of a circle between two points. These two points divide the circle into two arcs:
- Minor Arc: The shorter of the two arcs. Its measure is always less than 180 degrees.
- Major Arc: The longer of the two arcs. Its measure is always more than 180 degrees.
6. Why must the radius of a circle be perpendicular to the tangent at the point of tangency?
This is a fundamental theorem. The radius represents the shortest possible distance from the center of the circle to any point on the tangent line. In geometry, the shortest distance from a point to a line is always the perpendicular distance. Since the tangent only touches the circle at one point (the point of tangency), the radius connecting the center to this point must be perpendicular to the tangent line. Any other line from the center to the tangent would be longer and would lie outside the circle.
7. How does the property 'equal chords of a circle are equidistant from the centre' help in solving geometric problems?
This property creates a direct relationship between the length of a chord and its distance from the center. Its importance in problem-solving is significant:
- Finding Unknowns: If you know two chords in the same circle are equal, you can immediately conclude their perpendicular distances from the center are also equal, which helps find unknown lengths.
- Proving Congruency: This property is often used to prove that triangles formed by radii and chords are congruent, which is a key step in more complex geometric proofs.
- Symmetry: It reinforces the symmetric nature of a circle, showing that identical parts (chords of the same length) are positioned identically with respect to the center.
8. Is the diameter the only line of symmetry in a circle? Explain your reasoning.
No, the diameter is not the only line of symmetry; rather, any diameter is a line of symmetry. A line of symmetry is a line that divides a shape into two identical, mirror-image halves. Since a circle can have an infinite number of lines passing through its center to opposite points on the circumference, it has an infinite number of diameters. Each one of these diameters perfectly divides the circle, meaning a circle has an infinite number of axes of symmetry.
9. What is the importance of 'concyclic points' in understanding circle theorems?
Concyclic points are points that all lie on the circumference of the same circle. Their importance is foundational to many advanced circle theorems, particularly those in the CBSE Class 9 and 10 syllabus. For instance, the concept of a cyclic quadrilateral (a four-sided figure with all four vertices being concyclic) has special properties, such as its opposite angles summing to 180 degrees. Understanding concyclic points is essential for proving and applying theorems related to angles subtended by the same arc at the circumference and at the center.
10. How does the property 'the angle in a semicircle is a right angle' relate to other circle properties?
This property is a direct consequence of another major theorem: 'the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle'. In a semicircle, the arc is defined by the diameter. This diameter subtends a straight angle (180 degrees) at the center. Therefore, the angle it subtends at any point on the circumference must be exactly half of 180 degrees, which is 90 degrees. This elegantly connects the concepts of diameter, central angle, and inscribed angles.
