An isosceles trapezium (or trapezoid) is a special type of trapezoid where the non-parallel sides (legs) are equal in length. Calculating the perimeter of an isosceles trapezium is straightforward using the formula that sums the lengths of all its sides.
Formula to Find the Perimeter of an Isosceles Trapezium
The perimeter \( P \) of an isosceles trapezium ABCD, where \( AD \) and \( BC \) are the equal sides, can be calculated using the following formula:
\[ P = AB + CD + 2 \cdot AD \]
Alternatively:
\[ P = AB + CD + 2 \cdot BC \]
Where:
- \( P \) represents the perimeter of the trapezium.
- \( AB \) and \( CD \) are the lengths of the parallel sides (bases).
- \( AD \) and \( BC \) are the lengths of the non-parallel sides (legs).
Explanation of the Formula
The formula \( P = AB + CD + 2 \cdot AD \) or \( P = AB + CD + 2 \cdot BC \) is derived from the fact that the perimeter is the total length around the figure. Since \( AD \) and \( BC \) are equal in an isosceles trapezium, we multiply one of them by 2 to account for both sides.
Step-by-Step Calculation
Let's go through an example to illustrate how to use this formula.
Example:
Given an isosceles trapezium ABCD with the following side lengths:
- \( AB = 10 \) units
- \( CD = 14 \) units
- \( AD = BC = 7 \) units
We want to find the perimeter of the trapezium.
Step 1: Identify the Given Values
Given:
- \( AB = 10 \) units
- \( CD = 14 \) units
- \( AD = BC = 7 \) units
Step 2: Substitute the Values into the Formula
Using the formula \( P = AB + CD + 2 \cdot AD \):
\[ P = 10 + 14 + 2 \cdot 7 \]
Step 3: Calculate the Perimeter
First, multiply the length of the legs by 2:
\[ 2 \cdot 7 = 14 \]
Then sum the values:
\[ P = 10 + 14 + 14 \]
\[ P = 38 \]
Final Value
The perimeter of an isosceles trapezium ABCD with side lengths \( AB = 10 \) units, \( CD = 14 \) units, and \( AD = BC = 7 \) units is 38 units.
Using this simple formula, you can quickly determine the perimeter of any isosceles trapezium, making it a useful tool for various applications in geometry and real-life scenarios.
