Calculating the Area of a Trapezoid Given Base Angles and Base Lengths
In this article, we will show you how to calculate the area of a trapezoid given the base angles and the lengths of the bases. This step-by-step guide includes how to derive the height of the trapezoid using trigonometric functions and ultimately compute the area. Let's dive into the solution.
Introduction
In geometry, a trapezoid is a quadrilateral with at least one pair of parallel sides. Knowing the angles and lengths of the bases, we can calculate its area using specific formulas and trigonometric relationships. This article provides a detailed method to find the area of a trapezoid when given certain dimensions and angles.
Problem Statement
We are given a trapezoid with base angles of A 34° and B 48°, and the lengths of the upper and lower bases as b1 120 m and b2 220 m, respectively. The task is to find the area of the trapezoid.
Step-by-Step Solution
Find the height of the trapezoid:To find the height (h) of the trapezoid, we can drop perpendiculars from the top vertices to the line extending the bottom base. These perpendiculars will form right triangles with the bases. Define the variables and trigonometric relationships:
Let:
- h be the height of the trapezoid
- d1 be the horizontal distance from the top left vertex to the foot of the perpendicular dropped from it
- d2 be the horizontal distance from the top right vertex to the foot of the perpendicular dropped from it. Relate d1 and d2 to the difference in bases:
The total horizontal distance between the two bases can be expressed as:
d1 - d2 b2 - b1 220 m - 120 m 100 m
Using the angles:
For angle A 34°:
tan34° h / d1 rarr; d1 h / tan34°
For angle B 48°:
tan48° h / d2 rarr; d2 h / tan48°
Equating the above:
(h / tan34°) - (h / tan48°) 100 Calculate the height h:
Factoring out h: h [1 / tan34° - 1 / tan48°] 100
Using approximate values:
- tan34° ≈ 0.6745
- tan48° ≈ 1.1106
Substitute these values:
h [1 / 0.6745 - 1 / 1.1106] 100
Calculating the reciprocals:
1 / 0.6745 ≈ 1.4822 and 1 / 1.1106 ≈ 0.9003
Thus: h [1.4822 - 0.9003] 100
h 2.3825 100 rarr; h ≈ 100 / 2.3825 ≈ 41.9 m Calculate the area of the trapezoid:
Now we can use the height to find the area using the formula:
Area 1/2 (b1 b2)h
Substitute the values:
Area 1/2 (120 m 220 m) 41.9 m
1/2 340 m 41.9 m
170 m 41.9 m ≈ 7123 m2
Final Answer
The area of the trapezoid is approximately 7123 m2.
Conclusion
This method showcases the power of combining trigonometry with geometric formulas to solve complex problems. Understanding and applying these techniques can be invaluable in various fields, from engineering to construction.