A circle is a fundamental geometric shape that is defined as the set of all points in a plane that are equidistant from a fixed point, known as the center. This distance from the center to any point on the circle is called the radius. Circles are prevalent in various aspects of mathematics, science, and everyday life, making them essential to understanding. This complete guide delves into the properties of circles, exploring key concepts such as circumference, diameter, chords, tangents, and arcs. Whether you’re a student seeking to grasp basic principles or someone looking to deepen your knowledge, this comprehensive overview will provide you with a clear understanding of circles and their unique characteristics.

Table of Contents

## Definition of Circle

A circle is the collection of all the points in a plane that are at a given distance from a fixed point in the plane. This collection is all points that are in the plane. In this context, the fixed point is referred to as the center “O.” Listed below are some of the most significant terminologies that are used inside the circle.

Terms | Description |

Circumference | The total distance around the edge of a circle. It is like the perimeter of a circle. (C=2πr) |

Radius | The distance from the center of the circle to any point on its edge is called radius “r”. It is half the length of the diameter. |

Diameter | The distance across the circle, passing through the center. It is twice the length of the radius. (d=2r) |

Arc | A part of the circle’s edge or circumference. It is a curved line that is a section of the circle. |

Sector | A “slice” of the circle, like a piece of pie. It is the area between two radii and the arc connecting them. |

Chord | A straight line connecting two points on the circle’s edge. Unlike the diameter, it does not have to pass through the center. |

Tangent | A straight line that touches the circle at exactly one point. It is perpendicular to the radius at the point of contact. |

Secant | A straight line that intersects the circle at two points. It cuts through the circle, unlike the tangent that just touches it. |

## What is a Circle?

A circle is a simple closed curve in which all points are equidistant from a fixed central point. This fixed point is called the center of the circle, and the distance from the center to any point on the circle is known as the radius. The circle is a fundamental geometric shape characterized by its symmetry and uniformity.

Mathematically, a circle can be described by the equation ((x – h)² + (y – k)² = r²), where (h, k) are the coordinates of the center, and ( r ) is the radius. Circles are widely studied in geometry due to their unique properties and their appearance in various natural and man-made structures.

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## Properties of Circle

Here are some interesting properties that govern how their parts relate to each other. Here are some key properties of circles:

**Radius (r)**:

- The distance from the center of the circle to any point on its edge.
- All radii in a circle are equal.

**Diameter (d)**:

- The distance across the circle, passing through the center.
- It is twice the length of the radius: d=2r.
- It is the longest chord in a circle.

**Circumference (C)**:

- The total distance around the edge of the circle.
- Formula: C=2πr or C=πd.

**Area (A)**:

- The space is enclosed within the circle.
- Formula: A=πr².

**Chord**:

- A straight line connecting two points on the circle’s edge.
- Chords equidistant from the center are equal in length.

**Arc**:

- A part of the circumference between two points.
- It can be measured in degrees (angle subtended at the center) or length.

**Sector**:

- The area between two radii and the arc connecting them.
- Can be a minor sector (less than half the circle) or a major sector (more than half the circle).

**Segment**:

- The area between a chord and the arc it subtends.
- Can be a minor segment (smaller area) or a major segment (larger area).

**Tangent**:

- A straight line that touches the circle at exactly one point.
- Perpendicular to the radius at the point of contact.

**Secant**:

- A straight line that intersects the circle at two points.

**Central Angle**:

- An angle whose vertex is the center of the circle.
- Measures the angle subtended by an arc at the center.

**Inscribed Angle**:

- An angle formed by two chords in a circle that have a common endpoint.
- The measure of an inscribed angle is half the measure of the central angle subtending the same arc.

**Equal Arcs and Angles**:

- Arcs with equal lengths subtend equal angles at the center.
- Conversely, equal angles subtend equal arcs.

**Cyclic Quadrilateral**:

- A quadrilateral with all its vertices on the circle.
- Opposite angles of a cyclic quadrilateral sum up to 180∘180^\circ180∘.

**Concentric Circles**:

- Circles with the same center but different radii.

## Circle Formulas

Things around us are circle-shaped, like wheels, gears, planets, and coins. In addition to their everyday use, circles have interesting qualities that can be measured with math. This table lists the most important circle methods you’ll need to know.

Property | Formula |

Radius (r) | r = d / 2 |

Diameter (d) | d = 2r |

Circumference (C) | c = 2πr |

Area (A) | A = πr² |

Arc Length (L) | L = (θ/360) x 2πr |

Sector Area (A_s) | A_s = (θ/360°) x πr² |

Chord Length (c) | c=2rsin(θ/2) |

Segment Area (A_{seg}) | Aseg=As−1/2r²sinθ |

## Circle Solved Problems

Ready to put your circle knowledge to the test? Here are some numerical problems to solve, along with explanations to help you understand the concepts.

**Problem 1: Finding the Missing Radius**

**A bicycle wheel has a circumference of 210 cm. What is the radius of the wheel?**

**Solution:**

We know the circumference (c) is related to the radius (r) by the formula c = 2πr, where π (pi) is approximately 3.14.

Here, c = 210 cm. We need to find r.

Rearrange the formula to solve for r: r = c / (2π).

Plug in the value of c: r = 210 cm / (2 x 3.14) ≈ 33.73 cm.

**Problem 2: Calculating Area of a Sector**

**A cake is in the shape of a circle with a radius of 12 cm. If a slice has a central angle of 45 degrees, what is the area of the cake slice?**

**Solution:**

The area of a sector (A_s) can be calculated using the formula A_s = (θ/360°) x πr², where θ is the central angle and r is the radius.

Here, θ = 45° and r = 12 cm.

Plug in the values: A_s = (45°/360°) x 3.14 x (12 cm)² ≈ 21.2 cm².

**Problem 3: Missing Chord Length**

**A circle has a diameter of 20 cm. A chord is drawn inside the circle, and the distance between the center of the circle and the midpoint of the chord is 8 cm. Find the length of the chord.**

**Solution:**

- First, find the radius (r) by dividing the diameter (d) by 2: r = d/2 = 20 cm / 2 = 10 cm.
- Since the distance from the center to the midpoint of the chord is given, we can use the formula for chord length (applicable when the perpendicular bisects the chord): c = 2√(r² – d²), where c is the chord length.

Here, d (distance from center to midpoint) = 8 cm and r = 10 cm.

Plug in the values: c = 2√((10 cm)² – (8 cm)²) ≈ 6 cm.

**Problem 4: Area of a Circle Segment**

A circular pizza has a radius of 15 cm. A slice is cut out with a central angle of 120 degrees. What is the area of the remaining pizza (the segment)?

**Solution:**

- First, calculate the area of the whole pizza (circle) using A = πr², where A is the area and r is the radius. Here, A = 3.14 x (15 cm)² ≈ 706.5 cm².
- Next, calculate the area of the removed slice (sector) using A_s = (θ/360°) x πr², where A_s is the sector area, θ is the central angle (120°), and r is the radius (15 cm).

A_s ≈ (120°/360°) x 3.14 x (15 cm)² ≈ 235.5 cm².

- Finally, the area of the remaining pizza segment is the whole pizza area minus the removed slice area: Segment Area = Whole Pizza Area – Sector Area.

Segment Area ≈ 706.5 cm² – 235.5 cm² ≈ 471 cm².

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## FAQs

**How many properties does a circle have?**

Three primary measurements to keep in mind are the circumference, which is the circle’s edge, the diameter, which is the distance between the circle’s ends, measured across the middle, and the radius, which is half of the diameter.

**Which defines a circle?**

A circle is just a round shape with no edges or points. By mathematics standards, it is a closed circle. It is always the same distance between the circle’s points and its center.

**What is a formula for a circle?**

The standard form of the equation of a circle is with (h, k) center and r radius is given by: (x-h)2 + (y-k)2 = r2.

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