Area and Circumference of a Circle

Subject: Compulsory Maths

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Overview

Area of a Circle

The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2.

Area and Circumference of a Circle

Area of a Circle

The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2.

Here is a way to find the formula for the area of a circle:

Cut a circle into equal sectors (16 in this example)

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Rearrange the 16 sectors like this:

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These sectors look like a rectangular region but not exactly so. The length of this rectangle will be equal to half of the circumference and breadth equal to the radius of the circle.

We know that:

Circumference = 2 × π × radius

And so the width is about:

Half the Circumference = π × radius

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Now we just multply the width by the height to find the area of the rectangle:

Area = (π × radius) × (radius)

= π × radius2

Hence, we get

Area of the circle = \(\frac{1}{2}\) circumference x radius

= \(\frac{1}{2}\) x 2\(\pi\)r x r2

= \(\pi\)r2

Circumference of a Circle

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Verification:

Draw three circles of different radii. Measure the diameter of each one of them with the help of scale andfill teh table given below:

Circle Radius(r) Diameter(2r) Circumference(c) \(\frac{Circumference}{Diameter}\)=\(\frac{c}{2r}\)
(i)        
(ii)        
(iii)        

The ratio is denoted bycalled pi (\(\pi\))

Here,

\(\pi\) = 3.14(nearly)

= \(\frac{22}{7}\) (nearly)

Thus,

\(\frac{Circumference}{Diameter}\) = \(\pi\)

or, \(\frac{c}{2r}\) = \(\pi\)

or, c = 2\(\pi\)r

\(\therefore\) Circumference of a circle(c) = 2\(\pi\)r

Things to remember
  • Area of circle 
  • Circumference of circle 
  • It includes every relationship which established among the people.
  • There can be more than one community in a society. Community smaller than society.
  • It is a network of social relationships which cannot see or touched.
  • common interests and common objectives are not necessary for society.
Questions and Answers

Solution:

Area of a circle = 154 cm2

So, \(\pi\)r2= 154

or, r2= \(\frac{154}{\pi}\)

or, r2= \(\frac{154\times7}{22}\)

or, r2= 49

or, r= \(\sqrt{49}\)

\(\therefore\) r =7cm

Soution:

The circumference of the circle is given by

c =2\(\pi\)r

=2\(\times\)\(\frac{22}{7}\)\(\times\)10.5

=66 cm

Thus, circumference=66 cm

Solution:

We know that, c=2\(\pi\)r

or, 88=2\(\times\)\(\frac{22}{7}\)\(\times\)r

or, r=\(\frac{88\times7}{2\times22}\)\(\times\)r

or, r=\(\frac{88 x 7 }{2 x 22}\)

\(\therefore\) r=14cm

Thus, radius=14cm

Solutions:

Note that, in 1 revolution the car covers a distance equal to the circumference of the wheel.

Now, the diameter of the wheel=63 cm

Therefore, radius(r)=\(\frac{63}{2}\)cm

Circumference of the wheel= 2\(\pi\)r

=2\(\times\)\(\frac{22}{7}\)\(\times\)\(\frac{63}{2}\)

=198cm

=1.98m

Here, the distance covered in 1 revolution=1.98 m

Distance covered in 1000 revolutions=1.98\(\times\)1000

=1980m

Solution:

Circumference= 44 cm

So, 2\(\pi\)r=44

or, r=\(\frac{44}{2\pi}\)

or, r=\(\frac{44\times7}{2\times22}\)

\(\therefore\) r=7cm

Area of the circle=\(\pi\)r2

=\(\frac{22}{7}\)\(\times\)7 \(\times\)7

=154cm2

Solution:

Circumference of circle = 2πr
= 2 × 22/7 × 7
= 44cm

Area of circle = πr2
= 22/7 × 7 × 7 cm2
= 154cm2

Solution:

Given, Diameter (d) = 6 yards

π = 3.14

Now,

Circumference of a circle (c) = πd

= 3.14\(\times\)6

= 18.84yards

Solution:

Given, Diameter(d) = 6inch

Now,

Radius (r) = \(\frac{1}{2}\)d

=\(\frac{1}{2}\)6

= 3inches

Solution:

Given, Radius(r) = 1mile

π = 3.14

Now,

Circumference of circle (c) = 2πr

= 2\(\times\)3.14\(\times\)1

= 6.28miles

Solution:

Given, radius (r) = 3millimeters

Now,

Diameter (d) = 2r

= 2\(\times\)3

= 6millimeters

Solution:

Given, Circumference of a circle (c) = 6.28miles

π = 3.14

Now,

Radius (r) = \(\frac{c}{π}\)

= \(\frac{6.28}{3.14}\)

= 2miles

Solution:

Given, Circumference of a circle (c) = 6.28miles

π = 3.14

Now,

Radius (r) = \(\frac{c}{2π}\)

= \(\frac{3.14}{2\times3.14}\)

= 5miles

Solution:

Given, Radius (r) = 4millimeters

π = 3.14

Now,

Circumference (c) = 2 π r

= 2\(\times\)3.14\(\times\)4

= 25.12millimeters

Solution:

Given, Radius (r) = 10millimeters

π = 3.14

Now,

Circumference (c) = 2πr

= 2\(\times\)3.14\(\times\)10

= 62.8millimeters

Solution:

Given, Radius (r) = 3millimeters

π = 3.14

Now,

Circumference (c) = 2πr

= 2\(\times\)3.14\(\times\)3

= 18.84millimeters

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