Calculating the Perpendicular Distance Between Parallel Sides of a Trapezium
In geometry, a trapezium, also known as a trapezoid in some regions, is a quadrilateral with at least one pair of parallel sides. The parallel sides are known as the bases, and the non-parallel sides are called the legs. Calculating the perpendicular distance between these bases, given the areas and sums of the bases, is an important task in geometry. This article explores how to find the perpendicular distance using the area formula for a trapezium, providing detailed examples and explanations.
Understanding the Trapezium Area Formula
The area of a trapezium is a fundamental concept in geometry, and it can be calculated by using the formula:
A ? × (b1 b2) × h
where, A is the area, b1 and b2 are the lengths of the parallel bases, and h is the perpendicular distance (height) between these bases.
Example Problem
Let's consider a real-life example where the area of a trapezium is given as 63 square cm, and the sum of the lengths of its two parallel sides is 22.5 cm. We need to find the perpendicular distance (height) between these two parallel sides.
Step-by-Step Solution
Given Information: Area (A) 63 cm2. Sum of the lengths of the parallel sides (b1 b2) 22.5 cm.
Using the Area Formula: Substitute the given values into the formula:
63 ? × 22.5 × h
Simplify the multiplication on the right side:
63 11.25 × h
Isolating the Height (h): Solve for h by dividing both sides of the equation by 11.25:
h 63 / 11.25
h 5.6 cm
Therefore, the perpendicular distance (height) between the two parallel sides is 5.6 cm.
Alternative Methods
There are multiple ways to approach this problem:
The formula directly provided in the problem:
A (b1 b2) / 2 × h
Substituting the given values:
63 (22.5 / 2) × h
63 11.25 × h
solving for h, we get h 5.6 cm.
Using the trapezium's area as a function:
63 (22.5 × h) / 2
simplifying, we get:
126 22.5 × h
solving for h, we get h 5.6 cm.
Key Concepts in Geometry
The concept of the trapezium area is widely used in various practical applications, such as in architecture, construction, and design. Understanding the relationship between the area, bases, and height is essential for many geometric problems and real-world scenarios.
Conclusion
By applying the formula for the area of a trapezium, one can easily find the perpendicular distance between two parallel sides given the total area and the sum of the bases. The problem presented here demonstrates the general method and steps involved in solving such geometric problems, providing a clear and systematic approach to understanding this concept.
Related Keywords
This article covers important keywords in geometry such as:
trapezium area parallel sides perpendicular distance geometric calculations