Perimeter and Area of a Circle
Distance covered by travelling once around a circle is its perimeter, usually called its circumference.
circumference of a circle bears a constant ratio with its diameter. This constant ratio is denoted by the Greek letter π (read as ‘pi’).
circumference = π × diameter
= π × 2r (where r is the radius of the circle)
= 2πr
where π =$22/7$=3.14
area of a circle = πr2,
where r is the radius of the circle.
Example: The cost of ploughing a circular field at the rate of ₹ 0.50 per $m^2$ is
₹ 1925. The field is to be fenced at the rate of ₹ 24 per $m$.
Find the cost of fencing the field (Take π= $22/7$ ).
Total cost of ploughing the field = total area × cost per area
i.e 1925 = A × 0.5
i.e A = $1925/0.5$ = 3850
=> $πr^2 = 3850$
=> $22/7 × r^2 $ = $3850$
=> $r^2 $ = ${3850 × 7}/22$
=> $r=35$ $m$
Perimeter of the field = $2πr$
= $2×22/7×35$
= $2×22/7×35$
= $220$ $m$
Cost of fencing = ₹ 24 per $m$
Total cost of fencing = 24 × 220
= ₹ 5280
Example: The cost of fencing a circular field at the rate of ₹ 24 per metre is
₹ 5280. The field is to be ploughed at the rate of ₹ 0.50 per $m^2$. Find the cost of ploughing the field (Take π= $22/7$ ).
Step for find the cost of ploughing the field.
Step 1 Since the cost of ploughing is to be find, find the area of the circular field.
Step 2 Since the area of circular field is given as $πr^2$, and radius of the field
is not known, so first find the area of the field from the known data.
Let Black color denotes the length of the circular fence and grey color denotes the area of field.
Given: cost of fencing per meter = ₹ 24
Total cost of fencing a circular field = ₹ 5280
Length of the fence (in metres) = ${Total cost}/{Rate}$ = $5280/24$ = 220 $m$
So, circumference of the field = 220 $m$
Therefore, if $r$ is the radius of the field, then
$2πr$ = 220
2 × $22/7$ × $r$ = 220
$r$= ${220 × 7}/{2 × 22 }$= 35
i.e., radius of the field is 35 m.
Therefore, area of the field = $πr^2$ = $22/7$× 35 × 35 $m^2$ =22 × 5 × 35 $m^2$
Now, cost of ploughing 1 $m^2$ of the field = ₹ 0.50
So, total cost of ploughing the field = ₹ 22 × 5 × 35 × 0.50 = ₹ 1925