Acute angled triangle

A triangle is a two-dimensional geometrical shape having three sides and three angles. Based on the length of sides and the measure of interior angles, a triangle is categorized into three categories namely,

1. Based on the length of sides
   1. Equilateral triangle: A triangle having all its sides of equal length is termed as an equilateral triangle. Each of the interior angles of equilateral triangle are equal and of the measure 60°.
   2. Isosceles triangle: A triangle that contains two of its sides with equal length and two angles equal is termed an isosceles triangle. The angles corresponding to the equal sides are equal.
   3. Scalene triangle: A triangle with all its sides of different length is termed as a scalene triangle. The angles corresponding to each side are of different measure.
2. Based on the measure of angle
   1. Right triangle or right-angled triangle: A triangle which has one of its angles measuring 90° is called as a right-angled triangle. The other two angles are acute angles i.e. less than 90°.
   2. Acute angled triangle: A triangle having all its angles measuring less than 90° is called an acute angled triangle.
   3. Obtuse angled triangle: a triangle which has one of its angles measuring more than 90° is called an obtuse angled triangle. The other two angles are less than 90°.

Acute angled triangle

An acute angled triangle is the one which has all its interior angles less than 90° i.e. all the three angles of an acute angled triangle are acute. For example,

In any traingle ABC, ∠ABC = 40°

      ∠ACB = 75°

      ∠CAB = 65°

Here, all the angles of the given triangle are less than 90o. Therefore, ABC is an acute angled triangle.

Properties of acute angled triangle

1. The measure of all the interior angles is less than 90°.
2. The angle opposite to the largest side is the largest angle and the converse is also true.
3. The angle opposite to the smallest side is the smallest and the converse is also true.
4. The square of the largest side is equal to the sum of the square of the other two sides. Mathematically,
   c² = a² + b²
   where c is the largest side and a and b are the smallest.
5. An equilateral triangle is a type of acute-angled triangle.
6. Unlike an obtuse triangle, the orthocenter and circumcenter of an acute triangle are present in the interior of the triangle.

Perimeter of an acute-angled triangle

The perimeter of an acute-angled triangle is obtained by calculating the sum of its sides. For an acute triangle, let the measure of its sides be ‘a’ units, b units and c units then,

perimeter = a + b + c

Area of an acute-angled triangle

The area of an acute triangle can be calculated by two methods.

Method 1: By using the area of a triangle

We know that the area of a triangle in a two-dimensional plane is

Area = ½ . base. height

For a triangle of base b and height h

Area = ½ . b. h

Method 2: By using Heron’s formula

By Heron’s formula, we have known,

Area = √[s . (s-a) . (s-b) . (s-c)]

Where ‘s’ is semi perimeter calculated as

S = (a + b + c)/2