Area And Fencing Cost Calculation For A Triangular Park A Practical Guide

Hey guys! Ever wondered how much grass you'd need to plant in a triangular park or how much fencing it would take to go around it? Let's dive into a fun math problem that tackles just that! We'll be helping our gardener friend, Dhania, figure out the area of her triangular park and the cost of fencing it. So, grab your thinking caps, and let's get started!

Understanding the Triangular Park

Triangular park dimensions are the key to solving this problem. Imagine a park shaped like a triangle, which we'll call ABC. This park has sides of 120 meters, 80 meters, and 50 meters. Dhania needs to put a fence around the entire park, and she also wants to plant grass inside. Our mission is to figure out how much area she needs to plant with grass and how much the fencing will cost. This involves a couple of key steps: first, calculating the area of the triangle, and second, finding the perimeter to determine the length of the fence. Don't worry, we'll break it down step by step.

To find the area, we'll use a cool formula called Heron's formula, which is perfect for finding the area of a triangle when you know the lengths of all three sides. This formula is especially handy when the triangle isn't a right-angled triangle, making the traditional base-height method a bit tricky. Once we have the area, we'll know exactly how much space Dhania needs to cover with grass. Next, we'll calculate the perimeter by simply adding up the lengths of all three sides. The perimeter will tell us the total length of the fence needed. With the length of the fence and the cost per meter, we can easily figure out the total cost of fencing. So, let's get into the nitty-gritty of the calculations and help Dhania get her park looking beautiful!

Calculating the Area Using Heron's Formula

Heron's formula for triangle area is a game-changer when you know the lengths of all three sides but not the height. This formula might look a bit intimidating at first, but trust me, it's super useful! It states that the area of a triangle can be calculated using the formula: Area = √[s(s-a)(s-b)(s-c)], where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter. The semi-perimeter is simply half the perimeter of the triangle, calculated as s = (a + b + c) / 2. Now that we've got the formula down, let's apply it to our triangular park!

In our case, the sides of the park are 120 meters, 80 meters, and 50 meters. So, a = 120 m, b = 80 m, and c = 50 m. First, we need to calculate the semi-perimeter, s. Plugging in the values, we get s = (120 + 80 + 50) / 2 = 250 / 2 = 125 meters. Now that we have the semi-perimeter, we can plug all the values into Heron's formula. The area of the park is √[125(125-120)(125-80)(125-50)]. Let's simplify this step by step. The expression inside the square root becomes √[125 * 5 * 45 * 75]. Multiplying these numbers together, we get √[2109375]. Taking the square root of this value gives us approximately 1452.37 square meters. So, Dhania needs to plant grass in an area of about 1452.37 square meters. Isn't it amazing how a single formula can help us find such an important measurement?

Determining the Fencing Cost

Fencing cost calculation involves finding the perimeter of the triangular park and then multiplying it by the cost per meter of the barbed wire. The perimeter of any shape is the sum of the lengths of all its sides. In our case, the sides of the triangular park are 120 meters, 80 meters, and 50 meters. So, the perimeter is simply 120 + 80 + 50 = 250 meters. This means Dhania needs 250 meters of fencing to go all around the park. Now, let's consider the cost of the barbed wire.

The problem states that the cost of fencing is ₹20 per meter. To find the total cost, we multiply the total length of the fence needed (250 meters) by the cost per meter (₹20). So, the total cost of fencing is 250 * 20 = ₹5000. Therefore, Dhania will need to spend ₹5000 to fence the entire park with barbed wire. This calculation is straightforward but crucial for budgeting and planning any fencing project. By finding the perimeter and knowing the cost per unit, we can easily determine the total expense. This is a practical application of basic geometry in everyday situations, helping us make informed decisions about our projects and expenses.

Summary: Area and Fencing Cost for Dhania's Park

Park area and fencing cost are two crucial factors Dhania needs to consider for her triangular park. We've successfully calculated both, so let's recap the results. First, we found the area of the park using Heron's formula. We determined that Dhania needs to plant grass in an area of approximately 1452.37 square meters. This calculation involved finding the semi-perimeter of the triangle and then applying Heron's formula, which is a powerful tool for finding the area when you know all three sides.

Next, we tackled the fencing cost. We calculated the perimeter of the park by adding the lengths of all three sides, which gave us a total of 250 meters. Knowing that the cost of fencing is ₹20 per meter, we multiplied the total length of the fence by the cost per meter to find the total fencing cost. The result was ₹5000. So, Dhania will need to spend ₹5000 to fence her triangular park. These calculations provide Dhania with the information she needs to plan her project effectively. She now knows how much grass to buy and how much to budget for fencing. This example demonstrates how mathematical concepts can be applied to real-world situations, helping us solve practical problems. Whether it's gardening, construction, or any other project, understanding basic geometry and formulas can save time, money, and effort.

Practical Applications of Area and Perimeter Calculations

Real-world area and perimeter applications extend far beyond just gardening projects like Dhania's park. These calculations are fundamental in various fields, from construction and landscaping to interior design and urban planning. Understanding how to calculate area and perimeter allows us to make informed decisions, estimate costs, and plan projects effectively. Let's explore some practical examples where these concepts come into play.

In construction, calculating the area is essential for determining the amount of flooring, roofing, or siding material needed for a building. Accurate area calculations ensure that you purchase the correct amount of materials, minimizing waste and saving money. Similarly, perimeter calculations are crucial for estimating the amount of fencing required for a property or the length of baseboards needed for a room. In landscaping, area calculations help determine the amount of sod, mulch, or fertilizer needed for a garden or lawn. Perimeter calculations are used to estimate the length of edging or fencing required to define garden beds or property lines. Interior designers use area calculations to determine the amount of paint needed for walls, the square footage of carpet or tiles, and the size of furniture that will fit comfortably in a room. Perimeter calculations are helpful for planning the layout of furniture and ensuring adequate spacing for movement. Urban planners use area and perimeter calculations to design parks, roads, and buildings within a city. Accurate measurements are essential for optimizing space, ensuring efficient use of resources, and creating functional and aesthetically pleasing environments. From small home improvement projects to large-scale urban developments, the ability to calculate area and perimeter is a valuable skill that empowers us to tackle a wide range of challenges.

Conclusion

Mastering area and perimeter calculations is not just about solving math problems; it's about equipping ourselves with practical tools that we can use in everyday life. We've seen how Dhania can use these calculations to plan her gardening project, and we've explored numerous other real-world applications, from construction and landscaping to interior design and urban planning. Whether you're figuring out how much paint to buy for a room, estimating the cost of fencing a yard, or designing a new park, the ability to calculate area and perimeter is essential.

The problem of Dhania's triangular park perfectly illustrates how math concepts translate into tangible solutions. By using Heron's formula, we were able to find the area of the park, which helped Dhania determine how much grass she needed. Calculating the perimeter allowed us to estimate the cost of fencing, ensuring that Dhania could budget her project effectively. These skills are not limited to specific professions or situations; they are valuable for anyone who wants to make informed decisions and manage resources effectively. So, next time you encounter a situation that requires calculating area or perimeter, remember the formulas and the practical applications we've discussed. With a little bit of math, you can tackle a wide range of challenges and bring your projects to life!