1. Parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Because of this property, opposite sides are also equal in length.
If one pair of opposite sides has length \(a\) and the other pair has length \(b\), then the boundary of the parallelogram goes in the pattern: \(a \rightarrow b \rightarrow a \rightarrow b\).
1.1. Definition of Perimeter
The perimeter of a parallelogram is the total distance around its boundary. Since opposite sides are equal, we add the lengths of both pairs of sides.
2. Perimeter Formula
Let the lengths of the two pairs of opposite sides be \(a\) and \(b\). The perimeter is:
\( P = 2(a + b) \)
2.1. Working of the Perimeter Formula
A parallelogram has:
- Two sides of length \(a\)
- Two sides of length \(b\)
So, adding all the side lengths:
\( P = a + b + a + b = 2a + 2b = 2(a + b) \)
3. Examples
Here are some simple examples to understand how the formula is applied.
3.1. Example 1
A parallelogram has side lengths \(a = 8\,\text{cm}\) and \(b = 5\,\text{cm}\). Its perimeter is:
\( P = 2(a + b) = 2(8 + 5) = 2 \times 13 = 26\,\text{cm} \)
3.2. Example 2
A parallelogram-shaped field has sides \(a = 40\,\text{m}\) and \(b = 25\,\text{m}\). The boundary length is:
\( P = 2(a + b) = 2(40 + 25) = 2 \times 65 = 130\,\text{m} \)