In geometry, a triangle is the most important shape that falls under two-dimensional figures. A triangle is a three-sided closed polygon that has three vertices. It has three angles formed between each of the two sides. The sum of all the three angles of a triangle is always equal to 180 degrees. The sum of two sides of a triangle is always greater than the third side. These are all the properties of triangles. But the two important properties of the triangle are area and perimeter.
The area of a triangle is the region occupied by it, in a 2d space. The formula to find the area is:
Area of triangle = ½ x base x height
Where base and height are the dimensions of the triangle. It is measured in squared units.
The area for the three different types of triangles is also different. The three different types are scalene, isosceles and equilateral. A scalene triangle has all three sides unequal, an isosceles triangle has two of the three sides equal and an equilateral triangle has all its sides equal. The above-given formula is applicable to all the three triangles.
There is another formula to calculate the area of a triangle when all three sides of the triangle are known to us, which is called the Heron’s formula. This formula is used when we don’t know the altitude of the triangle, but we do know the length of its three sides. The formula is given by:
Area = √[s(s−a)(s−b)(s−c)]
Where ‘s’ is the semi-perimeter of the triangle, which is equal to, s = (a+b+c)/2 and a,b,c are the three sides of the triangle.
Also, there are other three types of triangles based on angles. They are acute-angled triangles, obtuse-angled triangles and right-angled triangles. The areas for these three triangles could be determined if we know the length of base and altitude of triangles.
We can find the area of any triangle if the length of the two sides and the angle between them are known to us. The formula then becomes:
Area = (½) × ab × Sin θ
Where a and b are the lengths of two sides and θ is the angle between them.
Now let us see, what is a perimeter of a triangle? A perimeter is the length of the outer boundary of a two-dimensional shape. Therefore, the perimeter of a triangle is the sum of the length of all the three sides. If a, b and c are the three sides of a triangle, then the perimeter is:
Perimeter = a + b + c
Since the scalene triangle has all three sides unequal, therefore the perimeter formula remains the same. But for isosceles and equilateral triangles, the perimeter formulas are different.
Perimeter of Isosceles Triangle = 2a + c, since a = b
Perimeter of equilateral triangle = 3a, since a = b = c
The three sides of a triangle are not always known to us. Therefore, in geometry, we are introduced with Pythagoras theorem. As per this theorem, the square of the hypotenuse of a right-angle triangle is equal to the sum of the square of base-side and square of perpendicular sides. The hypotenuse, base and perpendicular are called here Pythagorean triples. With the help of this theorem, we can find the sides of the triangle. But it is applicable only for a right triangle. Therefore, we cannot use it for other types of triangle.