**Table of Contents**

1. | Introduction |

2. | What is a circle? |

3. | Terminologies |

4. | Formulae |

5. | Properties of circle |

6. | Examples |

7. | Summary |

8. | FAQs |

14^{th} December 2020

Reading Time: 8 Minutes

**Introduction**

Circles are familiar to us from time immemorial. Our ancestors must have noticed the natural circular shapes of the Moon or Sun in the sky. One of the earliest inventions of humankind - the wheel is proof of our ancestors’ knowledge about circles.

The name ‘Circle’ is derived from a Greek word meaning ‘ring’ or ‘hoop’.

In this article, we will learn all about circles and their properties.

**Also read:**

**What is a circle?**

A circle is a curved plane figure. Every point on the circle is equidistant from a particular point, called the center of the circle. The curve drawn connecting all these points is called the circumference.

**Equation of a circle**

A circle can be drawn on a Cartesian plane using the general equation given below

r = radius of the circle

x = x coordinate

y = y coordinate

**Terminologies**

In this section, we will see the definition of some commonly used terminologies with regards to circles.

**Radius**

Radius is the distance of any point on the circle’s circumference from its center. The plural form is radius is radii.

**Chord**

A circles chord is a line passing through two points on a circle’s circumference is called a Chord.

** Diameter**

The diameter of a circle is the length of the chord passing through the center of the circle.

** Arc**

Any portion (part) of a circumference is called an Arc.

**Sector**

An area inside a circle formed by two radii and an arc is called a Sector.

**Segment**

An area of a circle formed by a chord and an arc is called the Segment.

**Concentric Circles**

Two or more circles that have the same center point are called Concentric circles.

**Annulus**

An area/region formed between two concentric circles is called Annulus. It is flat shaped, like a ring.

**Secant**

A line that touches two or more points on an arc/circumference of a circle is called Secant.

**Tangent**

A line that touches the arc/circle at one point is called a Tangent line.

**Formulae**

In this section, we will see the list of important formulae pertaining to any circle.

**Area of a circle**

Area can be defined as the size of any surface.

For a circle, area is given by the product of π and the square of the circle’s radius.

**Formula**

π = mathematical constant (pi) = 22/7

r = radius of the circle

**Circumference of a circle**

Circumference of a circle is the length of the boundary of the circle. It can also be considered as the perimeter of the circle.

**Formula**

π = mathematical constant (pi) = 22/7

r = radius of the circle

**Length of Arc**

As seen previously in the article, an arc is a section (part) of the circumference.

**Arc Length Formula **

ϴ = angle made at the center by the arc

r = radius of the circle

**Area of Sector**

The area of the sector of a circle can be derived from the formula of Area of circle. Let us see how.

Consider the picture shown below,

The sector shown in this picture makes an angle ϴ at the center.

We know that the area of a circle is π × r^{2} (when the angle made by the circle is 2 π). So, the area of the region which makes an angle ϴ is

**Formula**

ϴ = angle made by the arc (when measured in radians)

r = radius of the circle

**Length of Chord**

Length of a chord can be calculated if the angle made by the chord at the center and the value of radius are known.

**Formula**

ϴ = angle made at the center (measured in radians)

r = radius

**Area of Segment**

As seen earlier, segment is the region formed by the chord and the arc covered by the segment, as shown below.

**Formula**

r = radius

ϴ = angle made by the chord at the center (measured in radians)

A very important point to note is that the angle in a circle is sometimes given in terms of radians (as seen in the Formulae section) or can be given in terms of degrees too. Any formula can be modified to accommodate the angle given in degrees by a simple conversion formula given below.

**Properties of Circle**

In this section, we will see a list of important properties associated with Circle.

- Two circles can be called Congruent if they have the same radius.
- Two arcs of a circle can be called Congruent if both the arcs make the same angle at the center.
- If any 2 arcs of a circle are congruent, then we must know that their chords are equal and vice versa
- Equal chords are always equidistant from the center of the circle
- The distance from the center of the circle to the longest chord is zero (The longest chord is the diameter and it passes through the center)
- The perpendicular bisector of a chord must pass through the center of the circle
- When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their center points
- The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc on the circumference
- Tangents drawn right at the points where the diameter meets the circle are parallel to each other

**Examples**

**1. Find the area of a circle with diameter as 14 cm?**

**Answer:**

We have already learned the formula for calculating the area of the circle.

Area (A) = π × r × r

We know the value of π to be 22/7

We can find the value of radius (r) from the value of the diameter given in the question.

If diameter = 14 cm, then radius must be half of diameter which is 7 cm.

Now let us just substitute these values in the formula and find the answer.

Area = (22/7) × 7 × 7

Area = 154 cm^{2}

Area = 154 cm^{2} |

**2. Find the diameter of a circle, whose area is 204 cm ^{2}? **

**Find the arc length of a circle, if the radius of the circle is 5 cm and the angle made by the arc at the center is 110 degrees?**

**Answer:**

The formula for finding the length of the arc (when angle is measured in degrees) is

Length (l) = ϴ × r × π/180

We know ϴ to be 110^{o}

radius is 5 cm

π is 22/7

So, substituting these values in the formula, we get

Length (l) = 110 × 5 × (22/7) / 180

Arc length (l) = 9.6 cm

Arc length (l) = 9.6 cm |

**3. Find the angle (in radians), if the circumference of the circle is 54 cm and the arc length is 18 cm?**

**Find the length of the chord in a circle with radius 8 cm and central angle of 110 degrees?**

**Answer:**

Formula for finding the length of the chord when the angle is given in degrees is

Chord length (cl) = 2 × r × sin (ϴ/2)

We know ϴ to be 110^{o}

radius (r) = 8 cm

So, substituting these values in the formula, we get

Chord length = 2 × 8 × sin (110/2)

Chord length = 13.1 cm

Chord length = 13.1 cm |

**4. Find the length of the chord of a circle with a diameter of 4 m and a central angle of 90 degrees?**

**Find the area of a segment of a circle whose central angle is 100 degrees and radius is 4 cm?**

**Answer:**

The formula to find the area of segment while angle is given in degrees) is

Area of segment = (1/2) × (ϴ - sin ϴ) × r^{2}

We know r = 4 cm

And ϴ = 100

So, substituting these values in the formula, we get

Area = (1/2) × (100 – sin 100) × 16

Area of segment = 792 cm^{2}

Area of segment = 792 cm^{2} |

**5. Find the area of the segment in a circle whose radius is 7 m and the central angle is 50 degrees?**

**Answer:**

Calculate the missing angle in the given circle using angle properties

From the given figure, we see that the arc AB subtends an angle of 85^{o} at the center.

The same arc AB makes an angle x at the point C on the circumference.

We know that the angle made at the circumference is half of the angle made by the same arc at the center (as seen earlier in this article).

So, angle x = 85/2

x = 42.5^{o}

x = 42.5^{o} |

**6. Find the value of angle a in this cyclic quadrilateral figure**

**Answer:**

From the picture, we know that ∠ACB is 90^{o} (Angle made by the lines connecting the ends of a diameter on the circle is always 90^{o}.)

We also know that the angle ∠ABC is 125^{o}

We know that the opposite angles in a cyclic quadrilateral is 180^{o}

So, angle ∠ADC should be 180 – 125 = 55^{o}

Now, we know two of the angles inside the triangle △ADC. We can find the third angle (a) using the rule – Sum of the angles in a triangle is 180^{o}.

So, ∠a = 180 – (90 + 55)

∠a = 55^{o}

∠a = 55^{o} |

Also read,

**Summary**

In this article we learnt about circles, some of the common mathematical terminologies used in association with circles, various formulae used to calculate and measure different aspects of the circle and also some useful properties of circles.

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**Frequently Asked Questions**

**What is a cyclic quadrilateral?**

A cyclic quadrilateral is a quadrilateral that has all its 4 vertices (corners) on the circumference of the same circle.

The circle is called the Circumscribed circle and the vertices are concyclic.

The opposite angles in a cyclic quadrilateral are supplementary (they add up to 180^{o} )

**What is the major arc of a circle?**

Consider two points on the circumference of a circle. The larger arc which connects the two points is called the Major arc.

**What is the minor arc of a circle?**

Consider two points on the circumference of a circle. The smaller arc which connects the two points is called the Minor arc.

**What is a secant line?**

A line that touches two or more points on an arc/circumference of a circle is called the Secant line.

**What is a tangent?**

A line that touches the arc/circle at one point is called a Tangent line.

**What are concentric circles?**

Two or more circles which have the same center point are called Concentric circles.

*Written by Gayathri Sivasubramanian, Cuemath Teacher*