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Areas Related to Circles

Perimeter and Area of Circle and Semi-Circle

If you want to fence a plot, how would you calculate how much fencing is required? Or if your parents want to put a new carpet in your room, how much carpet you need to buy? Clueless about the concepts of Area and Perimeter of Circle and Semi-Circle? Don’t stress you will be getting all your answers in this section.

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Circle

Going by the basic definition it is a closed plane geometric shape. Now getting into 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 circle. The radius of a circle has a major role for calculation of perimeter and area of circle and semi-circle. Most famous examples of a circle in a line are pizza, chapatti, wheel etc.

Perimeter of Circle

A wheel

Terminologies Associated with a Circle

Radius

The line joining centre and any point on the boundary of the circle or, the distance between any points on the periphery of the circle to the centre is called radius of the circle. It is generally denoted by ‘R’ or ‘r’

Perimeter of Circle

Radius of a circle

Diameter

Line segment double the span of the radius of the circle passing through the centre of the circle, with its ends lying on the boundary of a circle is called diameter of the circle. Length of the diameter of the circle, denoted by ‘D’ = 2R.

Perimeter of circle

Cicumference=Perimetre of circle and Diametre of circle

Area and Perimeter of Circle and Semi-Circle

Perimeter

The perimeter of a plane 2-D figure is the length of the outer periphery of the figure. The figure can be any regular or irregular polygon; it does not have to be a regular geometric figure. So now if you want to fence a plot, how would you calculate how much fencing is required? Yes, by simply calculating the perimeter of the plot, you would be able to calculate the amount of fencing required. The perimeter of a circle can also be specified as Circumference of the circle.

 

Derivation of Perimeter of Circle

As a circle is not a regular figure so obviously we cannot use a ruler for calculating the circumference of the circle. To calculate the circumference of the circle we can follow the following steps:

  • Step 1: Take out some amount of thread; trace the path of the circular plane object with the thread.
  • Step 2: Then measure the length of the thread required for tracing the entire boundary of the circular plane with a ruler. The measured length will be the circumference of the circle.

This gives another definition of the circle that it the total length of the rope which wraps around its boundary perfectly will be equal to its circumference.

Browse more Topics under Areas Related To Circles

Geometrical calculation of circumference(Perimeter of circle)

  • Step 1: Use ruler or measuring tape to measure the radius of the circular plane.
  • Step 2: Put the measured value of the radius in the following formula and calculate:

The formula for the circumference of a circle:

C = 2πR = πD

where, C= circumference and π= It is constant pronounced as “pi” with the value of 22/7 or 3.14

Learn more about Area and Perimeter Related to Arcs of Circle here in detail.

Circumference/Perimeter of  Semi-Circle

Semi-circle is exactly half of a complete circle then its circumference or perimeter should also be half of the circumference/perimeter of circle of a complete circle, it is so but a semicircle has some part of its periphery in form of straight line so circumference or more precisely in case of semi-circle, perimeter is:   πR + 2R

Perimeter of circle

2R= Diametre of Semi-Circle

Area of Circle

Area of any geometrical figure is the space occupied by it on a two-dimensional plane. Now, what is the area of a circle? Well, the area of a circle is the space occupied by it with a certain radius on a two-dimensional plane. So, if your parents want to put new carpet in your circular room, how much carpet you need to buy? Again, it will be equal to the area of the room.

Understand the Theorems Related to Chord of the Circle here.

Area of Circle = πR2

perimeter of circle

Derivation

As we have already seen the rectangular method of derivation of the area of a circle. This time we shall be following the triangular approach.

Triangular Approach

  • Step 1: Draw a circle. For the construction of a circle draw it with a compass or follow the following procedure: Take a paper pin fixed centre of the circle to be constructed fix the pin over it then tie a thread on the pointed lower tip of the pin tie other end the thread to the tip of the pen or pencil you are using for construction after this trace out the complete circular path with the pen or pencil around the fixed centre.
  • Step 2: Draw concentric rings of equal width inside the constructed circle in the following manner:
Perimeter of circle

Concentric Rings of Circle

  • Step 3: Make a cut with scissors along the radius of the circle following which cut down the concentric rings drawn such that they form straight strips.
  • Step 4: Arrange the strips in form of a pyramid with the longest outermost strip at the bottom and the shortest innermost one at the top.
  • Step 5: Now the triangle of pyramid formed will have the lowermost strip of length equal to the circumference of the circle and the height will be equal to the radius of the circle.

Area of Triangle form= ½(Height) x (Base)

Here, Height= Radius of circle= R and Base = Circumference of the circle=2πR. Therefore,

Area= ½(R) x (2R) = πR= πD2/4

Area of Semi-Circle

As the area of a complete circle is πR2 then going by the unitary method the area of a semi-circle will be πR2 /2.

Understand the concept of the Unitary method here.

Solved Examples for You

Question 1: For a vehicle having wheels of radius 24cm find the distance covered by it in one complete revolution of wheels.

Answer: Distance covered by wheel in one complete revolution = circumference of wheel=Perimeter of Circle= 2πR

Here, R=24cm

Distance covered = 2x(22/7)x24

= 66×3 =198 cm

Question 2: What is the formula to get the circumference of a circle?

Answer: The circumference of a circle is proportional to its diameter and its radius. However, the formula of the circumference is \(\pi\) × d (diameter of a circle). When you multiply the value of pi (22/7 or 3.14) with the diameter of the circle you get the circumference of the circle.

Question 3: State the formula of the perimeter of a semi-circle?

Answer: The perimeter of a circle is \(\pi\) × d. Here ‘d’ is the diameter of the circle. Hence, the perimeter of a circle is half of the that of the circle that is ½ \(\pi\) × d.

Question 4: How to find the area of a circle?

Answer: For finding the area of a circle with the radius, firstly, square the radius, or multiply it by itself. Next, multiply the squared radius by pi to get the area. Furthermore, find the area with the diameter, simply divide the diameter by 2, moreover, put the value in the radius formula, and solve it as before.

Question 5: What is the arc length formula?

Answer: The formula of arc length is L = r × Θ. Here Θ is the central angle, r is the radius.

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