Calculating the Height of a Trapezoid Given Its Area
Understanding the properties and formulas associated with geometric shapes is essential in both academic and practical applications. A trapezoid is one such shape, and it's important to know how to find its height when given its area and the lengths of its bases. In this article, we'll walk through a step-by-step process to solve such a problem, using a specific example.
Formulas and Definitions
A trapezoid is a quadrilateral with one pair of parallel sides, known as the bases. The area of a trapezoid can be determined using the following formula:
Area of a Trapezoid
Variable definitions:
A Area (in square feet or square meters) h Height (in feet or meters) A Lower Base (in feet or meters) B Upper Base (in feet or meters)The formula for the area of a trapezoid is:
[text{Area} frac{1}{2} times (A B) times h]Solving for the Height
Let's use the given problem to illustrate the process:
Example Problem
Given: Area (A) 164 square feet, Lower Base (A) 12.8 feet, Upper Base (B) 20 feet Find: Height (h)Step 1: Substitute the known values into the area formula.
[text{164} frac{1}{2} times (12.8 20) times h]Step 2: Simplify the equation by adding the lengths of the bases.
[text{164} frac{1}{2} times 32.8 times h]Step 3: Multiply 32.8 by (frac{1}{2}).
[text{164} 16.4 times h]Step 4: Isolate the height by dividing both sides by 16.4.
[text{h} frac{164}{16.4}]Step 5: Calculate the result.
[text{h} approx 10 text{ feet}]Alternative Method
Another way to solve for the height involves manipulating the area equation:
[text{A} frac{1}{2} times (A B) times h]Step 1: Multiply both sides of the equation by 2.
[text{2A} (A B) times h]Step 2: Divide both sides by the sum of the bases (A B).
[text{h} frac{2A}{A B}]Step 3: Substitute the given values into this equation.
[text{h} frac{2 times 164}{12.8 20}]Step 4: Simplify the expression.
[text{h} frac{328}{32.8}]Step 5: Calculate the result.
[text{h} approx 10 text{ feet}]Conclusion
By following a systematic approach, we were able to determine the height of the trapezoid given its area and the lengths of its bases. This method not only provides a clear solution but also reinforces the importance of understanding geometric formulas. Whether you are a math student, a professional in a related field, or just someone interested in geometry, these techniques can be valuable in various scenarios.