The perimeter of a circular lawn is 1232 m. There is 7 m wide path around the lawn. The area (in m 2) of the path is: Take \(\left(\pi=\frac{22}{7}\right)\)
8778
Calculating the Area of the Path Around a Circular Lawn
This problem asks us to find the area of a path surrounding a circular lawn. We are given the perimeter of the circular lawn and the width of the path. To solve this, we first need to find the radius of the lawn using its perimeter. Then, we can find the radius of the larger circle that includes the lawn and the path. Finally, we calculate the area of both circles and subtract the area of the lawn from the area of the larger circle to find the area of the path.
Finding the Radius of the Circular Lawn
The perimeter of a circle is given by the formula \( P = 2\pi r \), where \( P \) is the perimeter and \( r \) is the radius. We are given that the perimeter of the circular lawn is 1232 m and we should use \( \pi = \frac{22}{7} \). Let \( r_1 \) be the radius of the circular lawn.
So, we have: \( 2\pi r_1 = 1232 \) \( 2 \times \frac{22}{7} \times r_1 = 1232 \) \( \frac{44}{7} r_1 = 1232 \)
To find \( r_1 \), we multiply both sides by \( \frac{7}{44} \): \( r_1 = \frac{1232 \times 7}{44} \) We can simplify this calculation. Divide 1232 by 44: \( 1232 \div 44 = (1232 \div 4) \div 11 = 308 \div 11 = 28 \) So, \( r_1 = 28 \times 7 \) \( r_1 = 196 \) m.
The radius of the circular lawn is 196 m.
Determining the Radius of the Outer Circle
A 7 m wide path is around the circular lawn. This means the outer circle, which includes the lawn and the path, has a radius that is the radius of the lawn plus the width of the path. Let \( r_2 \) be the radius of the outer circle.
\( r_2 = \text{radius of lawn} + \text{width of path} \) \( r_2 = r_1 + 7 \) \( r_2 = 196 + 7 \) \( r_2 = 203 \) m.
The radius of the outer circle is 203 m.
Calculating the Area of the Path
The path is the region between the outer circle and the inner circle (the lawn). This shape is called an annulus. The area of the path is the area of the outer circle minus the area of the inner circle. The area of a circle is given by the formula \( A = \pi r^2 \).
Area of the path \( A_{path} = \text{Area of outer circle} - \text{Area of inner circle} \) \( A_{path} = \pi r_2^2 - \pi r_1^2 \) We can factor out \( \pi \): \( A_{path} = \pi (r_2^2 - r_1^2) \)
We know \( r_1 = 196 \) m and \( r_2 = 203 \) m, and \( \pi = \frac{22}{7} \). \( A_{path} = \frac{22}{7} (203^2 - 196^2) \)
We can use the difference of squares formula, \( a^2 - b^2 = (a-b)(a+b) \), to simplify the calculation: \( 203^2 - 196^2 = (203 - 196)(203 + 196) \) \( 203 - 196 = 7 \) \( 203 + 196 = 399 \) So, \( 203^2 - 196^2 = 7 \times 399 \).
Now substitute this back into the area of the path formula: \( A_{path} = \frac{22}{7} \times (7 \times 399) \) We can cancel out the 7 in the denominator and the numerator: \( A_{path} = 22 \times 399 \)
Finally, perform the multiplication: \( 22 \times 399 = 22 \times (400 - 1) = (22 \times 400) - (22 \times 1) \) \( 22 \times 400 = 8800 \) \( 22 \times 1 = 22 \) \( A_{path} = 8800 - 22 = 8778 \) m\(^2\).
The area of the path around the circular lawn is 8778 m\(^2\).
| Concept | Formula | Description |
|---|---|---|
| Circumference (Perimeter) of a Circle | \( C = 2\pi r \) or \( C = \pi d \) | Distance around the circle. \( r \) is radius, \( d \) is diameter. |
| Area of a Circle | \( A = \pi r^2 \) | Space enclosed by the circle. \( r \) is radius. |
| Area of an Annulus (Path) | \( A_{path} = \pi (r_2^2 - r_1^2) \) | Area between two concentric circles. \( r_1 \) is inner radius, \( r_2 \) is outer radius. |
Additional Information on Circular Geometry
Understanding the geometry of circles is fundamental in solving problems like finding the area of a path. Here are a few related concepts:
- Annulus: The ring-shaped region between two concentric circles (circles with the same center but different radii). The path in this problem is an example of an annulus.
- Units: Always pay attention to the units given in the problem. Perimeter is a length and is measured in meters (m), while area is a measure of two-dimensional space and is measured in square meters (m\(^2\)).
- Value of \( \pi \): The problem specifies using \( \pi = \frac{22}{7} \). In other problems, you might be asked to use \( \pi \approx 3.14 \) or a more precise value. Using the specified value is important for matching the expected answer.
- Difference of Squares: The algebraic identity \( a^2 - b^2 = (a-b)(a+b) \) is very useful in geometry problems involving the difference of areas of squares or circles, as shown in this solution.
Important Questions from Plane Figures
If length of a rectangle is increased to its three times and breadth is decreased to its half, then the ratio of the area of given rectangle to the area of new rectangle is:
The width of the path around a square field is 4.5 m and its area is 105.75 m 2. Find the cost of fencing the field at the rate of Rs. 100 per meter.
What is the area of the square (in cm 2) whose vertices lie on a circle of radius 5 cm?
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
If the radius of a circle is decreased by 11% then the total decrease in the area of the circle is given as:
