Perimeter and Areas Related to Circle

Perimeter and area of a circle

PI-

• The number π is a mathematical constant.
• It is defined as the ratio of a circle’s circumference to its diameter. And its value is
• From the above definition, we can derive the following equation for
• Circumference/Diameter = π,
• The great Indian mathematician Aryabhata (C.E. 476 – 550) gave an approximate value of π. He stated that π=22/7 = 3.1416 approx

Circumference of a circle

• The perimeter of a circle is the distance covered by going around its boundary once.
• The perimeter of a circle has a special name: Circumference, which is π times the diameter
• which is given by the formula 2πr.

You may be having doubt how it is 2πr?

Here is proof:

As we know π= Circumference/Diameter —–(1)

Diameter= 2*radius —–(2)

From 1 and 2 we can write,

Π=Circumferance/2Xradius —- (3)

From above equation 3, we get,

Circumference or perimeter= 2πr.

Area of a Circle

Area of a circle is π r², where π = or 22/7 ≈ 3.14

where r is the radius of the circle.

You may be having doubt how it is?

Proof of Area of a circle.

Consider the following diagram,

 

You can see that the shape in Fig. 2 is nearly a rectangle with

Length = 1/2*2π r —-(1)

and

breadth = r —-(2)

This suggests that the area of the rectangle = length*breath

Area =  2π r × r =2π r²

Let’s Solve some Example problems based on circumference and perimeter:

Q.1.   The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has a circumference equal to the sum of the circumferences of the two circles.

Sol. We have, r₁ = 19 cm

                 r₂ = 9 cm

        ∴ Circumference of circle-I = 2π r₁ = 2π (19) cm

        Circumference of circle-II = 2π r₁ = 2π (19) cm

Sum of the circumferences of circle-I and circle-II = 2π (19) + 2π (9)

= 2π (19 + 9) cm

= 2π (28) cm

  Let R be the radius of the circle-III.

        ∴ Circumference of circle-III = 2π R

        According to the condition,

                2π R = 2π (28)

        

    Thus, the radius of the new circle = 28 cm.

Q.2.   The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having an area equal to the sum of the areas of the two circles.

Sol. We have,

         Radius of circle-I, r₁ = 8 cm

         ∴ Area of circle-I = πr₁ ² = π(8)² cm²

                 Area of circle-II = πr_2² = π(6)2 cm²

        Let the area of the circle-III be R

        ∴ Area of circle-III = πr₂

        Now, according to the condition,

                πr₁² + πr₂² = πr₂

        i.e. π(8)² + π(6)² = πr₂

        ⇒ π(8² + 6²) = πr₂

        ⇒ 8² + 6² = r₂

        ⇒ 64 + 36 = r₂

        ⇒ 100 = r₂

        ⇒ 10² = r₂ ⇒ R = 10

        Thus, the radius of the new circle = 10 cm.