What are the 7 Important Properties of a Circle?
The concept of circles in Maths plays a key role in geometry and is widely used in both real-life situations and exams. From wheels and coins to graphic design and physics, circles appear everywhere, making it an essential topic for school students.
What Is a Circle in Maths?
A circle in Maths is defined as the set of all points in a plane that are at a fixed distance, called the radius, from a given fixed point called the centre. You’ll find this concept applied in coordinate geometry, trigonometry, and practical measurement tasks. The circle is a 2-dimensional, round figure, and all its points are equidistant from the centre.
Key Formulas for Circles in Maths
Here are the most important formulas related to circles in Maths:
| Quantity | Formula | Description |
|---|---|---|
| Radius (r) | Distance from center to any point on circle | Basic measurement in a circle |
| Diameter (d) | d = 2r | Longest chord, passes through centre |
| Circumference (C) | C = 2πr or πd | Perimeter/outer length of a circle |
| Area (A) | A = πr² | Space occupied by the circle |
| Equation (center at (h, k)) | (x - h)² + (y - k)² = r² | For coordinate geometry |
Parts of a Circle and Properties
Circles in Maths include important parts like:
- Centre – fixed point (O)
- Radius – line from centre to boundary
- Diameter – chord passing through centre
- Chord – joins any two points on circle
- Arc – a curved part of circumference
- Sector – region between two radii and arc
- Segment – region between chord and arc
- Secant – a line cutting the circle at two points
- Tangent – a line touching the circle at only one point
Key properties include:
- All radii in a circle are equal.
- Diameter is the longest chord.
- Circles with the same radius are congruent.
- The perpendicular from centre bisects a chord.
- Equal chords are equidistant from centre.
Step-by-Step Illustration: Finding Area and Circumference
Let’s solve a simple example for better understanding.
1. Given: Radius (r) = 7 cm2. Formula for area: \( A = \pi r^2 \)
3. Substitute the values: \( A = \frac{22}{7} \times 7 \times 7 = 154 \) cm²
4. Formula for circumference: \( C = 2\pi r \)
5. Substitute the values: \( C = 2 \times \frac{22}{7} \times 7 = 44 \) cm
So, the area is 154 cm² and circumference is 44 cm.
Examples of Circles in Real Life
You see circles everywhere! Some real-world examples are:
- Clock face
- Wheels of a bicycle/car
- Coins
- Bangles and rings
- Plates, buttons, and CDs
Speed Trick or Vedic Shortcut: Remembering Formulas
Here’s a quick way to remember circle formulas: The word “P-A-C” stands for Perimeter (Circumference) = 2πr, Area = πr², and Centre at (h, k). Making a short sound (like ‘pack’) helps students recall during exams.
Many students also draw a simple diagram and label radius, diameter, and centre for instant recall.
Try These Yourself
- Draw a circle with radius 5 cm using a compass.
- If the circumference is 31.4 cm, find the radius.
- Which is the longest chord in a circle?
- List three objects around you in the shape of a circle.
Frequent Errors and Misunderstandings
- Mixing up circumference and diameter formulas.
- Applying the area formula wrongly for sphere or semicircle.
- Forgetting the equation of the circle in coordinate geometry.
Relation to Other Concepts
The idea of circles in Maths is closely related to Parts of Circle and Area of a Circle. Mastering circles will also help you with topics like Circle Theorems (angles, tangents), and understanding Difference Between Circle and Sphere for exams and higher classes.
Classroom Tip
Quickly memorise the basics with the CARe (Centre, Area, Radius) mnemonic—write down C for Centre, A for Area, R for Radius. Many Vedantu teachers use such tricks and diagrams in online classes to help students visualise and speed up revision for circles in Maths.
We explored circles in Maths—from definition, types, formulas, solved examples, common mistakes, and connections to other chapters. Keep practising with Vedantu for more easy explanations, worked-out solutions, and exam tips on all maths concepts.
FAQs on Circles – Definition, Formulas, Properties & Examples
1. What is a circle in Maths?
A circle in Maths is a two-dimensional geometric shape defined as a set of points equidistant from a fixed point called the center. The fixed distance from the center to any point on the circle is called the radius.
2. What are the basic properties of a circle?
Key properties of a circle include:
• All points on the circle are equidistant from the center.
• The diameter, a chord passing through the center, is twice the radius.
• A chord connects any two points on the circle.
• A tangent is a line that touches the circle at exactly one point.
• The radius drawn perpendicular to a chord bisects the chord.
• Circles with equal radii are congruent.
• Circles of different sizes are similar.
3. How do you find the area and circumference of a circle?
The area of a circle is calculated using the formula: A = πr², where 'r' is the radius. The circumference (perimeter) is calculated using: C = 2πr or C = πd, where 'd' is the diameter.
4. What is the difference between a circle and a sphere?
A circle is a two-dimensional figure—a flat, round shape. A sphere is a three-dimensional figure—a round, solid object. A circle is like a slice through the middle of a sphere.
5. Give 3 real-life examples of circles.
Real-world examples of circles include:
• A wheel
• The sun (approximately)
• A coin
6. What is the equation of a circle?
The general equation of a circle with center (h, k) and radius r is: (x - h)² + (y - k)² = r²
7. How is the radius related to the diameter of a circle?
The diameter of a circle is always twice the length of its radius. The formula is: d = 2r.
8. What is a sector of a circle?
A sector is the region enclosed by two radii and the arc between them. Think of it like a slice of pie.
9. What is a segment of a circle?
A segment is the area enclosed by a chord and the arc it subtends. It's the area between a chord and the arc of the circle.
10. What is meant by the circumference of a circle?
The circumference is the total distance around the outside of the circle. It's the circle's perimeter.
11. How do you find the length of a chord in a circle?
The length of a chord depends on its distance from the center. If you know the radius (r) and the distance (d) from the center to the midpoint of the chord, you can use the Pythagorean theorem: chord length = 2√(r² - d²)
12. What are concentric circles?
Concentric circles are circles that share the same center but have different radii.
