1. Triangle
A triangle is a polygon with three sides and three vertices. The sides can be of equal or unequal lengths depending on the type of triangle.
The perimeter of a triangle is simply the total length of its three sides.
1.1. Definition of Perimeter
The perimeter of a triangle is defined as the sum of the lengths of all its three sides. If the sides are denoted by \(a\), \(b\), and \(c\), then the perimeter equals the total boundary length.
2. Perimeter Formula
If the sides of the triangle have lengths \(a\), \(b\), and \(c\), the perimeter is given by:
\( P = a + b + c \)
2.1. Working of the Formula
A triangle has exactly three sides. Therefore, to find how much distance you would cover by walking around the triangle, you add the lengths of all these sides together:
\( P = a + b + c \)
This works for any type of triangle—scalene, isosceles, or equilateral.
3. Special Cases of Triangles
Some triangles have special side properties. These make finding the perimeter even easier.
3.1. Equilateral Triangle
All three sides are equal. If each side is \(a\), then:
\( P = a + a + a = 3a \)
3.2. Isosceles Triangle
An isosceles triangle has two equal sides. If the equal sides are \(a\) and the base is \(b\), then:
\( P = a + a + b = 2a + b \)
4. Examples
Let’s look at a few examples to understand how to apply the formula.
4.1. Example 1
A triangle has sides \(a = 5\,\text{cm}\), \(b = 7\,\text{cm}\), and \(c = 6\,\text{cm}\). The perimeter is:
\( P = 5 + 7 + 6 = 18\,\text{cm} \)
4.2. Example 2
An equilateral triangle has side length \(a = 10\,\text{cm}\). Its perimeter is:
\( P = 3a = 3 \times 10 = 30\,\text{cm} \)