Triangles

Triangles are a three-sided polygon that consists of three edges and three vertices and the sum of internal angles of a triangle equal to 180o. Depending upon the sides and angles of a triangle, we can have different types of triangle, such as Isosceles triangle, Equilateral triangle,  acute triangle, right triangle, etc. The formula and properties of triangles are based on its sides and angles. In this article, we are also going to learn a brief description for determining the area of a triangle using direct formula and  Heron’s formula along with the properties of triangles.

Definition of Triangle

As we discussed in the introduction, a triangle is a type of polygon, which has three sides joint end to end and three angles, formed between two sides. This is one of the important parts of geometry. Some of the major concepts such as Pythagoras theorem and trigonometry are dependent on triangle properties. Let us learn now its types.

On the basis of length of the sides. triangles are classified into three categories:

  1. Scalene Triangle
  2. Isosceles Triangle
  3. Equilateral Triangle

On the basis of measurement of the angles, triangles are classified into three categories:

  1. Acute Angle Triangle
  2. Right Angle Triangle
  3. Obtuse Angle Triangle

Below given is a triangle having 3 sides and three edges numbered as 0,1,2.

Triangles

Types of Triangles

Scalene Triangle

This type of triangle has all its three edges of different length. Due to this, the three angles are also different from each other.Scalene Triangle

Isosceles Triangle

This type of triangle has two equal sides. Also, the two angles, opposite to the two equal sides, are also equal to each other. This fact also supports the famous isosceles triangle theorem.

Isosceles Triangle

Equilateral Triangle

This type of triangle has all three sides equal to each other. Due to this all the internal angles are of equal degrees ie. each of the angles is 60°

Equilateral Triangle

Acute Angle Triangle

This triangle has all of its angles less than 90°.

Acute Angle Triangle

Right Angle Triangle

This triangle has any of its one angles to be equal to 90°.

Right-Angled Triangle

Obtuse Angle Triangle

This triangle has any of its one angles more than 90°.

Obtuse Angled Triangle

Properties of Triangle

Each and every shape and figure in Maths have some properties which distinguish them from each other. Let us discuss here some of the properties of triangles.

  1. Sum of Angles of a Triangle is always 180 degrees.
  2. The exterior angles of a triangle always add up to 360 degrees.
  3. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.
  4. The shortest side is always opposite to the smallest interior angle. Similarly, the longest side is always opposite to the largest interior angle.


Area of a Triangle 

The area for a given triangle can be derived with the help of knowing the length of the base and height. Suppose a triangle with base ‘B’ and height ‘H’ is given to us.

Area Of Triangle

 

The area of a triangle is given by-

Area = 1/2 × Base × Height

Example

Question- Find the area of a triangle having base equal to 9 cm and height equal to 6 cm.

Solution-  We know that Area = 1/2 × Base × Height

= 1/2 × 9 × 6 cm2

= 27 cm2

 

Heron’s Formula To Find Area of Triangle

Sometimes it is difficult to calculate the height of a triangle, so we cannot use the direct formula.

Therefore Heron’s formula is used to calculate the area of a triangle by the means of sides of triangle.

First we need to calculate the semiperimeter (s).

s = 1/2 (a+b+c),                     (where a,b,c are the three sides of a triangle)

Now Area is given by

\(\bigtriangleup = \sqrt{s(s-a)(s-b)(s-c)}\)

Example

Question- Find the area of a triangle having sides 5,6 and 7 units length.

Solution-  Using Heron’s formula to find the area of a triangle-

Semiperimeter (s) = \( \frac{1}{2}(a+b+c)\)

s =  \( \frac{1}{2}(5+6+7)\)

s = \( 9 \)

Now Area \(\bigtriangleup = \sqrt{s(s-a)(s-b)(s-c)}\)

= \(\sqrt{9(9-5)(9-6)(9-7)}\)

= \(\sqrt{9\times 4 \times 3 \times 2}\)

\(= \sqrt{3 \times 3 \times 2 \times 2 \times 3 \times 2}\)

= 6√6 square units.

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Practise This Question

If ΔDEFΔBCA, then the part of Δ that correspond to E is

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