Calculate The Area Of A Circular Path Around A Garden

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This article provides a step-by-step solution for calculating the area of a circular path surrounding a circular garden. We will explore the concepts of area, radius, and the mathematical constant pi ($\pi$) to arrive at the answer. Understanding how to calculate such areas is not only useful for academic purposes but also for practical applications in landscaping, construction, and design. This article will guide you through the problem, breaking down each step to ensure a clear understanding of the process and the underlying mathematical principles. We will use the given values, apply the relevant formulas, and perform the calculations to determine the area of the path accurately. This will involve calculating the area of the larger circle (garden plus path) and subtracting the area of the garden itself. The difference will give us the area of the path alone. So, let's delve into the problem and discover the solution together.

Problem Statement

We are presented with a scenario involving a circular garden. This garden has a radius of 8 feet. Surrounding this garden is a circular path that is 3 feet wide. Our task is to determine the approximate area of the path alone. We are instructed to use 3.14 as the value for $\pi$. This problem involves geometric principles, specifically the calculation of the area of circles, and requires us to apply the formula for the area of a circle: $A = \pi r^2$, where A is the area and r is the radius. To find the area of the path, we will need to calculate the area of the larger circle (garden plus path) and subtract the area of the smaller circle (the garden). This will give us the area of the path alone. Let's proceed with the solution step by step.

H2: Solution

To find the area of the path, we need to calculate the area of the larger circle (garden plus path) and subtract the area of the garden. Let's break this down step by step:

Step 1: Determine the radius of the outer circle

The radius of the outer circle is the sum of the garden's radius and the width of the path. The garden has a radius of 8 feet, and the path is 3 feet wide. Therefore, the radius of the outer circle is:

router=8extfeet+3extfeet=11extfeetr_{outer} = 8 ext{ feet} + 3 ext{ feet} = 11 ext{ feet}

This step is crucial as it sets the foundation for calculating the area of the larger circle, which includes both the garden and the path. Understanding this combined radius is essential for accurately determining the total area before we can subtract the garden's area to find the path's area. This initial calculation allows us to proceed with the area calculations using the correct dimensions, ensuring the final result is precise and reflects the actual size of the path surrounding the garden. The accurate determination of the outer radius is paramount for the subsequent steps in solving the problem.

Step 2: Calculate the area of the outer circle

Now we will calculate the area of the outer circle using the formula for the area of a circle: $A = \pi r^2$. We know the radius of the outer circle is 11 feet, and we are using 3.14 for $\pi$. So,

Aouter=Ï€router2=3.14imes(11extfeet)2=3.14imes121extft2=379.94extft2A_{outer} = \pi r_{outer}^2 = 3.14 imes (11 ext{ feet})^2 = 3.14 imes 121 ext{ ft}^2 = 379.94 ext{ ft}^2

This step is a direct application of the formula for the area of a circle, where we substitute the known values of $\pi$ and the outer radius to find the total area enclosed by the outer circle. This area represents the combined space of the garden and the surrounding path. The accurate calculation of this area is vital because it serves as the starting point for finding the path's area. We will later subtract the garden's area from this total area to isolate the area of the path itself. Therefore, the precision in this step ensures the final answer accurately reflects the path's area.

Step 3: Calculate the area of the garden

The area of the garden can also be calculated using the formula for the area of a circle: $A = \pi r^2$. The garden has a radius of 8 feet, and we're using 3.14 for $\pi$. Therefore,

Agarden=Ï€rgarden2=3.14imes(8extfeet)2=3.14imes64extft2=200.96extft2A_{garden} = \pi r_{garden}^2 = 3.14 imes (8 ext{ feet})^2 = 3.14 imes 64 ext{ ft}^2 = 200.96 ext{ ft}^2

In this step, we apply the same formula for the area of a circle but use the garden's radius to calculate its area. This calculation is essential as we need to subtract the garden's area from the total area (garden plus path) to find the area of the path alone. The accuracy of this step is crucial for obtaining the correct final answer. By determining the garden's area precisely, we ensure that the subsequent subtraction will yield the exact area of the path, providing a clear solution to the problem. This step highlights the importance of using the correct dimensions and applying the formula consistently to achieve an accurate result.

Step 4: Calculate the area of the path

To find the area of the path alone, we subtract the area of the garden from the area of the outer circle:

Apath=Aouter−Agarden=379.94extft2−200.96extft2=178.98extft2A_{path} = A_{outer} - A_{garden} = 379.94 ext{ ft}^2 - 200.96 ext{ ft}^2 = 178.98 ext{ ft}^2

This step is the culmination of the previous calculations, where we subtract the garden's area from the total area (garden plus path) to isolate the area of the path itself. This subtraction provides the final answer to the problem, representing the approximate area of the path surrounding the garden. The accuracy of this step depends on the precision of the preceding area calculations. By correctly subtracting the garden's area from the outer circle's area, we arrive at the accurate measurement of the path's area, fulfilling the problem's objective. This step demonstrates the importance of careful calculation and logical progression in solving geometric problems.

H2: Final Answer

The approximate area of the path alone is 178.98 ft^2. Therefore, the correct answer is:

B. 178.98 ft^2

This final answer represents the solution to the problem, derived through a series of calculations involving the areas of circles. We first determined the radius of the outer circle, then calculated the areas of both the outer circle and the garden. Finally, we subtracted the garden's area from the outer circle's area to find the area of the path alone. This step-by-step approach ensures accuracy and clarity in problem-solving. The correct answer, 178.98 ft^2, provides the specific measurement of the path's area, addressing the original question posed in the problem statement. This result underscores the practical application of geometric principles in real-world scenarios, such as landscaping and design.

H2: Key Concepts Used

Area of a Circle

The area of a circle is a fundamental concept in geometry, defined as the space enclosed within the circle's boundary. The formula for the area of a circle is $A = \pi r^2$, where A represents the area, $\pi$ (pi) is a mathematical constant approximately equal to 3.14159, and r is the radius of the circle. The radius is the distance from the center of the circle to any point on its circumference. This formula is essential for calculating the size of circular regions in various applications, from determining the material needed for a circular tablecloth to calculating the cross-sectional area of a pipe. Understanding the relationship between the radius and the area allows for accurate measurements and planning in numerous fields, including engineering, construction, and design. The consistent application of this formula ensures precise results when dealing with circular shapes.

Radius

The radius is a crucial parameter of a circle, defined as the distance from the center of the circle to any point on its circumference. It is half the diameter, which is the distance across the circle through its center. The radius plays a fundamental role in various geometric calculations, particularly in determining the area and circumference of a circle. As seen in the formula $A = \pi r^2$, the area of a circle is directly related to the square of its radius, highlighting its significance. The accurate measurement or determination of the radius is essential for precise calculations involving circular shapes. In practical applications, the radius is used in fields such as engineering, architecture, and manufacturing to design and construct circular components and structures. Understanding the concept of the radius is vital for solving problems related to circles and their properties.

Pi ($\pi$)

Pi ($\pi$) is a fundamental mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation neither terminates nor repeats. The value of $\pi$ is approximately 3.14159, but it is often rounded to 3.14 for simpler calculations. Pi is a cornerstone of mathematics and is essential for calculating the area and circumference of circles, as well as the volume and surface area of spheres and other circular shapes. Its presence is ubiquitous in geometry, trigonometry, and calculus, making it one of the most important constants in mathematics. Pi's significance extends beyond theoretical mathematics into practical applications in physics, engineering, and computer science, where it is used in various formulas and algorithms. The accurate use of pi is crucial for precise calculations involving circular and spherical shapes.

H2: Practical Applications

Calculating the area of a circular path has numerous practical applications in real-world scenarios. One common application is in landscaping, where homeowners or professional landscapers need to determine the amount of material required for a path or border around a circular garden or feature. For example, if you're building a stone path around a circular flower bed, calculating the area helps you estimate the amount of stone needed, reducing waste and saving costs. In construction, this calculation is vital for designing and building circular structures, such as fountains, patios, or even circular buildings. Engineers and architects use these principles to ensure accurate material estimation and structural integrity. Furthermore, these calculations are essential in urban planning and park design, where circular paths and features are often incorporated for aesthetic and functional purposes. Understanding how to calculate the area of a circular path is therefore a valuable skill in various fields, contributing to efficient planning and resource management.

H2: Common Mistakes to Avoid

When calculating the area of a circular path, there are several common mistakes that students and individuals often make. One of the most frequent errors is forgetting to add the width of the path to the garden's radius when calculating the radius of the outer circle. This leads to an incorrect outer radius, which subsequently affects the area calculation. Another mistake is using the diameter instead of the radius in the area formula $A = \pi r^2$. It's crucial to remember that the formula requires the radius, which is half the diameter. A third common error is calculating the area of the outer circle and the garden separately but then failing to subtract the garden's area from the outer circle's area to find the path's area. This results in reporting the total area (garden plus path) instead of the path's area alone. Additionally, using an incorrect value for $\pi$ or rounding prematurely can introduce inaccuracies in the final result. To avoid these mistakes, it's essential to carefully read the problem, identify the correct values for the radii, use the correct formula, and follow the steps logically to ensure an accurate calculation of the circular path's area.