Calculating Trapezoid Height From Area And Bases

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Finding the height of a trapezoid when you know its area and base lengths is a common geometrical challenge. This article will walk you through the process, ensuring you understand each step involved. We'll tackle the specific problem: a trapezoid has an area of 70.55 square feet, with bases measuring 11.4 feet and 5.6 feet. What is the height of this trapezoid? Let’s dive in and solve it together!

Understanding the Trapezoid and Its Area

Before we jump into the calculation, let’s solidify our understanding of a trapezoid and its area formula. A trapezoid, also known as a trapezium, is a four-sided quadrilateral with at least one pair of parallel sides. These parallel sides are called the bases (often denoted as b₁ and b₂), and the perpendicular distance between them is the height (h). Understanding the unique characteristics of a trapezoid is crucial for accurately calculating its area and other properties.

The area of a trapezoid represents the two-dimensional space it occupies. To calculate the area of a trapezoid, we use the formula:

Area = (1/2) * (b₁ + b₂) * h

Where:

  • Area is the area of the trapezoid
  • b₁ and b₂ are the lengths of the two bases
  • h is the height (the perpendicular distance between the bases)

This formula is derived from the concept of averaging the lengths of the bases and multiplying by the height. Essentially, we're finding the average width of the trapezoid and multiplying it by the height, which mirrors how we find the area of a rectangle. The trapezoid area formula is a cornerstone of geometry, offering a precise method for determining the space enclosed within this unique quadrilateral. By understanding this formula, we unlock our ability to solve a myriad of problems, from simple textbook exercises to real-world applications in architecture and engineering. The area formula serves as a bridge between the trapezoid’s dimensions and its spatial extent, allowing us to quantify its size and make informed calculations. Remembering this formula is the first crucial step in unraveling the mystery of the trapezoid's height.

Applying the Formula to Find the Height

Now that we understand the formula for the area of a trapezoid, let's apply it to the problem at hand. We know the area is 70.55 square feet, and the lengths of the bases are 11.4 feet and 5.6 feet. Our goal is to find the height (h). To find the height of the trapezoid, we will rearrange the formula to solve for h. This involves a bit of algebraic manipulation, but it’s a straightforward process. We start with the formula:

Area = (1/2) * (b₁ + b₂) * h

We can plug in the values we know:

70.55 = (1/2) * (11.4 + 5.6) * h

First, let's simplify the expression inside the parentheses:

  1. 4 + 5.6 = 17

Now, substitute this back into the equation:

70.55 = (1/2) * 17 * h

Next, multiply (1/2) by 17:

  1. 5 * h = 70.55

To isolate h, we divide both sides of the equation by 8.5:

h = 70.55 / 8.5

Now, perform the division:

h = 8.3

Therefore, the height of the trapezoid is 8.3 feet. Understanding how to apply the formula in this way is crucial, not only for solving academic problems but also for practical applications in various fields. The ability to rearrange formulas and solve for unknown variables is a fundamental skill in mathematics and science. By mastering this technique, you can tackle a wide range of geometrical problems and apply these principles in real-world scenarios. This step-by-step approach to problem-solving not only yields the correct answer but also enhances your understanding of the underlying mathematical concepts. With a solid grasp of algebraic manipulation and the trapezoid area formula, you are well-equipped to solve similar problems with confidence and accuracy.

Step-by-Step Solution

To recap, let's break down the solution into clear, concise steps. This will help solidify your understanding and provide a roadmap for tackling similar problems in the future. Solving for the height of a trapezoid involves a systematic approach. By breaking down the process into manageable steps, we ensure accuracy and clarity in our solution. Let’s recap each step to solidify your understanding.

  1. Write down the formula for the area of a trapezoid:

    Area = (1/2) * (b₁ + b₂) * h

    This is the fundamental equation we'll be working with, so it's crucial to have it memorized or readily available.

  2. Substitute the given values into the formula:

    We know the area is 70.55 square feet, b₁ is 11.4 feet, and b₂ is 5.6 feet. Plugging these values in, we get:

    70.55 = (1/2) * (11.4 + 5.6) * h

    This step involves replacing the variables in the formula with the specific numbers provided in the problem.

  3. Simplify the expression inside the parentheses:

    Add the lengths of the bases:

    1. 4 + 5.6 = 17

    Our equation now looks like this:

    70.55 = (1/2) * 17 * h

    This simplification makes the equation easier to work with.

  4. Multiply (1/2) by the sum of the bases:

    1. 5 * h = 70.55

    This step further simplifies the equation, bringing us closer to isolating the variable h.

  5. Isolate h by dividing both sides of the equation by 8.5:

    h = 70.55 / 8.5

    Dividing both sides by the coefficient of h is a key algebraic technique for solving equations.

  6. Calculate the height:

    h = 8.3 feet

    This final step gives us the answer to the problem: the height of the trapezoid is 8.3 feet.

By following these steps, you can systematically solve for the height of any trapezoid, given its area and base lengths. Each step is logical and builds upon the previous one, ensuring a clear and accurate solution. Practicing this method will enhance your problem-solving skills and deepen your understanding of geometrical concepts. Remember, mathematics is a journey of understanding, and each step you take brings you closer to mastery.

Choosing the Correct Answer

Looking at the answer choices provided, we can confidently select the correct answer. Now that we have calculated the height of the trapezoid, it's essential to match our solution with the available options. The answer choices given are:

A. 4.15 ft B. 4.15 ft² C. 8.3 ft D. 8.3 ft²

We found that the height of the trapezoid is 8.3 feet. Therefore, the correct answer is C. 8.3 ft. It’s important to note the units. Since we are calculating a length (height), the unit is feet (ft), not square feet (ft²), which is a unit of area. This distinction is crucial in geometry and ensures that we are expressing our answers in the appropriate units. Selecting the correct answer also involves a careful review of the question and the units being asked for. Sometimes, students make mistakes by choosing an answer with the correct numerical value but the wrong units. Therefore, paying close attention to the units is a fundamental part of problem-solving in mathematics and science. Furthermore, understanding the difference between linear measurements (like height) and area measurements is key to avoiding errors. By correctly identifying the units and matching our calculated value with the provided options, we can confidently and accurately answer the question. This process reinforces the importance of both mathematical computation and careful analysis in problem-solving.

Conclusion: Mastering Trapezoid Calculations

In conclusion, we've successfully calculated the height of the trapezoid by understanding and applying the area formula. This exercise demonstrates the importance of understanding geometrical formulas and using them correctly. By mastering these concepts, you can confidently tackle similar problems in the future. Remember, the key to success in mathematics lies in understanding the fundamental principles and practicing their application. The ability to calculate the height of a trapezoid is a valuable skill, not only in academic settings but also in various real-world applications. From architecture and engineering to design and construction, understanding geometrical shapes and their properties is essential. This article has provided a step-by-step guide to solving this specific problem, but the underlying principles can be applied to a wide range of geometrical challenges. Continual practice and a solid understanding of the formulas are the keys to mastering these concepts. The journey of learning mathematics is one of continuous growth and discovery. Each problem solved adds to your understanding and builds your confidence. So, keep practicing, keep exploring, and keep pushing your boundaries. The world of geometry is vast and fascinating, and with the right tools and approach, you can unlock its many secrets. Embrace the challenge, and you'll find that mathematics can be both rewarding and empowering.