A triangle is the polygon bounded by least number of sides. As you all know a triangle has exactly three sides. Triangles are classified according to the angle measures or according to side lengths.

The classification done on the basis of number of congruent sides are as follows:
Scalene Triangle - In this type of triangle, the two sides are never congruent.
Isosceles Triangle - In this type of triangle, a minimum of two sides are congruent.
Equilateral Triangle - In this type of triangle, all the 3 sides are congruent.

Scalene triangles not only have distinct side lengths, but also distinct angle measures. The classification of Scalene triangles can be further refined considering the angle measures as well as, acute scalene, obtuse scalene and right scalene triangles.

Scalene Acute
Scalene Obtuse
Scalene Right
Scalene Acute Triangle:
Has three sides are of
different lengths and all
the three angles are acute.
Scalene Obtuse Triangle:
Has three sides are of
different lengths and one
obtuse angle at P.
Scalene Right Triangle:
Has three sides are of
different lengths and one
Right angle at R.

Scalene triangle is taken to be the default triangle, if no additional information is given.

The Perimeter of a triangle is found by adding the lengths of the three sides of the triangle.
Thus if the sides of the triangle are denoted by a, b and c then
Perimeter = (a + b + c) units.

For example, the perimeter of a triangle with side lengths 10 cm, 14 cm and 18 cm is,
P = (10 + 14 + 18) cm = 42 cm.
The formula used to calculate the area of a triangle is

Area = $\frac{1}{2}$ x base x height.

Length of any side can be taken as the base and the corresponding altitude as height.

Area of Triangle

Solved Example

Question: Find the area of a scalene triangle with base length = 6 cm and the corresponding height = 8 cm.
Solution:
 
Area of the triangle = $\frac{1}{2}$ x b x h = $\frac{1}{2}$ x 6 x 8 = 24

Thus area of the triangle is 24 cm2.

Heron's formula is another formula used to compute the area of a triangle when the lengths of the three sides are known.

Area of the triangle = $\sqrt{s(s-a)(s-b)(s-c)}$

where a, b and c are the side lengths of the triangle and s is known as the semi perimeter and s = $\frac{a+b+c}{2}$.
 


Solved Examples

Question 1: Find the perimeter and area of a triangle whose side lengths are given as 8 in, 12 in and 15 in.
Solution:
 
Perimeter of the triangle = (8 + 12 + 15)inches = 35 inches.

    The area of the triangle can be calculated using Heron's Formula

    a = 8, b = 12 and c = 15

    s = $\frac{a+b+c}{2}$ = $\frac{8+12+15}{2}$ = 17.5

    Area of the triangle = $\sqrt{s(s-a)(s-b)(s-c)}$

                                = $\sqrt{17.5(17.5-8)(17.5-12)(17.5-15)}$

                                = $\sqrt{17.5\times 9.5\times 5.5\times 2.5}$

                                ≈ 47.81

=> Area of the triangle is 47.81 Sq. inches (approx).
 

Question 2: The area of a triangle is 96 sq.ft.. If the length of the base is measured as 16 ft, find the length of the corresponding altitude.
Solution:
 
We use the area formula reversely to compute the height here.

    Area = $\frac{1}{2}$ b h = 96

               $\frac{1}{2}$ x 16 x h = 96

                                 h = $\frac{192}{16}$ = 12

    Hence, the length of the altitude corresponding to the given base is 12 ft.