Areas Related to Circles

Area and Perimeter Related to Arcs of a Circle

Yes! The pizza in your hand is a perfect example of a circle, didn’t bother to ever notice? Well in this section we shall be using it as our benchmark example to derive some interesting concepts of arcs of a circle. We will see in this section how to calculate area and perimeter of circle and arcs of a circle.

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Circle

Going by the basic definition, it is a closed plane geometric shape. Now getting into the technical jargon, a circle is a locus of a point moving in a plane in such a way that its distance from a fixed point is always constant. In layman terms, a round shape is often referred to as a circle. Point seeking attention here is that the fixed distance is the radius of the circleMost famous examples of a circle in a line are pizza, chapatti, wheel etc.

arcs of a circle

Basics of a circle

Circumference/Perimeter of a Circle

A perimeter of any geometrical figure is the length of the outer boundary of the shape. Similarly, in case of a circle, it’s perimeter is termed as circumference. On practical grounds, let’s take an example of a wheel, the distance covered by the wheel in one complete revolution will be the circumference of the wheel. 

The formula for the circumference of a circle is C= 2πR

where C= circumference,

R = Radius of the circle,

π = It is constant pronounced as “pi” with a value of 22/7 or 3.1416…

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Arcs of a Circle

Now that we are done with the circumference of the circle, what is an arc? Arc is a part of the circumference of a circle. If the length is zero, it will be merely a point on the boundary of the circle. And if it is of length, it will be the circumference of the circle i.e. an arc of length 2πR.

Length of the arc of a circle = (θ/360o ) x 2πR

θ=Angle subtended by an arc at the centre of the circle, measured in degrees. If you are working with angles measured in radians instead of in degrees, then go an extra mile converting it in degrees with the conversion factor: (180o/π) x θ

arcs of a circle

An arc of length L with a circle of radius R 

Derivation of Length of an Arc of a Circle

  • Step 1: Draw a circle with centre O and assume radius. Let it be R.
  • Step  2: Now, point to be noted here is that the circumference of circle i.e. arc of length 2πR subtends an angle of 360o at centre.
  • Step 3: Going by the unitary method an arc of length 2πR subtends an angle of 360o at the centre, Therefore; an arc subtending angle θ at the centre will be of length: (θ/360o ) x 2πR

Area of Arcs of a Circle or Sectors of a Circle

When we talk about area enclosed by arcs of a circle it is actually the space enclosed between the ends of the arc and the centre of the circle specifically area enclosed by an arc is the area of the sector of the circle. A sector of a circle is like a slice of a pizza where whole pizza is the complete circle.

arcs of a circle

Slice of a pizza same as a sector of a circle where pizza is the complete circle in the present case

Derivation for the Area of a Circle

  • Step 1: Consider a pizza of equal-sized slices.
  • Step 2: Arrange the slices such that they form a rectangle in the following manner:
arcs of a circle

Visualising area of a circle using Area of Rectangle

  • Step 3: Now, as we can see from the figure, the breadth of the rectangle is R, that is the radius of a circle and length is πR, which is half of the circumference of the circle – reason being that we have arranged slices in an inverted manner, alternatively half the number of slices will contribute to length on each side. Hence, the area of the circle comes out to be πR2.

Derivation for Area of an Arc

Following the unitary method the area of the arc subtending an angle of 360o at the centre, the angle subtended by a complete circle is πR2 then the arc suspending angle of θ will be:

Area enclosed by an arc of a circle or Area of a sector = (θ/360o ) x πR2

We have seen in this section how we are supposed to calculate area and perimeter of circle and arc. As we know mathematics is not a spectator sport so we also got through its application in some practical examples of area and perimeter related to circle and arc. Now I am sure you will be able to “calculate” the biggest slice of pizza for yourself.

Solved Example for You

Q: Find length of an arc of a circle of radius 14cm subtending an angle of 30o

Solution: Length of an arc of circle is  (θ/360o )x 2πR

θ= 30o; R=14 cm

Length of arc= (30o/360o )x 2(22/7)x14

= 88/12 cm =7.33 cm

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