We have already discussed about polygons in the Polygon and its Types chapter. There we learnt that polygon is a closed curve, which is made up of only line segments.

Triangle is such one type of polygon which has three sides or line segments. It is a polygon with least number of sides, in other words, a polygon with less than three sides does not exist.

Triangle is a closed figure which is formed by joining three line segments. It has 3 sides, 3 vertices and 3 angles.

Triangle is denoted by \(\triangle\), a delta symbol.

Lets, learn about triangle with the following figure.

Example of a triangle \(\triangle ABC\)

So, this is a triangle whose name is written as \(\triangle ABC\), where \(\triangle \) is a symbol of triangle and ABC is the name of triangle, which always includes three vertices of triangle. We can use any three vertices in any order to represent a triangle.

For example, ABC or BAC or CAB or ACB. Therefore, the triangle can also be written as \(\triangle BAC\) or \(\triangle CAB\) or \(\triangle ACB\).

As you know that triangle has three sides always. These sides are also called the line segments.

So, in \(\triangle ABC\), line segments AB, BC and CA are sides of the \(\triangle ABC\). These three sides of a triangle are written as \(\overline{AB}\), \(\overline{BC}\) and \(\overline{CA}\).

Vertex of a triangle is that point where any two sides of a triangle meet out or intersect.

In the \(\triangle ABC\) in above figure, sides AB and AC meet at point A. So, A is vertex of \(\triangle ABC\).

Similarly, sides BC and AB meet at point B. So B is a vertex of \(\triangle ABC\).

Also, sides AC and BC meet at point C. So C is a vertex of \(\triangle ABC\).

Therefore, A, B and C are the three vertices of \(\triangle ABC\).

Angles are formed at vertex of a triangle. It is a measurement of how slanted two lines are to each other. The angles are measured in degrees units.

Let’s understand angles in a triangle \(\triangle ABC\) from the above figure.

As we already discussed above that \(\triangle ABC\) has three vertices A, B and C.

Therefore, we can say angles are formed at vertices A, B and C. Angles are written as \(\angle ABC\), \(\angle BAC\) and \(\angle ACB\) or in a short form as \(\angle B\), \(\angle A\) and \(\angle C\) respectively.

In other words, we can say:

\(\angle ABC\) or \(\angle B\) is formed at vertex B of \(\triangle ABC\).

\(\angle BAC\) or \(\angle A\) is formed at vertex A of \(\triangle ABC\).

\(\angle ACB\) or \(\angle C\) is formed at vertex C of \(\triangle ABC\).

Triangle has many types depending upon the length of its sides and size of its angles.

A triangle is said to be equilateral triangle if all sides of a triangle are of equal length.

\(\triangle ABC\) is equilateral triangle because AB = BC = CA, where AB is length of side AB, BC is length of side BC and CA is length of side CA.

Equilateral triangle \(\triangle ABC\)

A triangle is said to be Isosceles triangle if any two sides of triangle are of equal length.

\(\triangle ABC\) is Isosceles triangle because AB = AC.

Isosceles triangle \(\triangle ABC\)

A triangle is said to be scalene triangle if all sides of triangle are of unequal length.

\(\triangle ABC\) is Scalene triangle because AB \(\neq\) BC \(\neq\) AC.

Scalene triangle \(\triangle ABC\)

A triangle is said to be acute angled triangle if each angle of a triangle is acute. Acute angle is that angle which is less than \(90^o\).

So, the \(\triangle ABC\) in figure is an acute angled triangle because all angles are less than \(90^o\). That is, \(\angle A\), \(\angle B\) and \(\angle C\) are all less than \(90^o\).

Acute angled triangle \(\triangle ABC\)

A triangle is said to be right angles triangle if one of its angle is right angle. Right angle is that angle whose measure is \(90^o\).

So, the \(\triangle ABC\) in figure is a right angled triangle because \(\angle C\) is equal to \(90^o\).

Right angled triangle \(\triangle ABC\)

A triangle is said to be obtuse angled triangle if one of its angles is obtuse angle. Obtuse angle is that angle which is greater than \(90^o\).

So, the \(\triangle ABC\) in figure is an obtuse angled triangle because \(\angle C\) is greater than \(90^o\).

Obtuse angled triangle \(\triangle ABC\)

Perimeter is the length of boundary around any closed figure.

The boundary of a triangle consists of its three sides. Therefore, sum of length of three sides of tringle is called its perimeter.

Units of perimeter measurement of a triangle is taken same as units of length of sides of the triangle.

The few examples of units of length of side of triangle are meter (m), centimeter (cm) etc.. So, the units of perimeter are also taken as meter (m), centimeter (cm) etc..

Perimeter of triangle \(\triangle ABC\)

In this \(\triangle ABC\), we have

length of AB = a

length of BC = b

and length of CA = c

So, perimeter of \(\triangle ABC\) is sum of length of sides AB, BC and CA

**i.e. perimeter of \(\triangle ABC\)** = \(a + b + c\)

Equilateral triangle \(\triangle ABC\) with sides length a

In equilateral triangle, where all sides are of equal length.

i.e. AB = a, BC = a and CA = a

So, **perimeter of equilateral \(\triangle ABC\)** = \(a + a + a = 3a\)

Isosceles triangle \(\triangle ABC\) with sides length a, b and a

In isosceles triangle, where any two sides are of equal length.

i.e. AB = a, BC = a and CA = b

So, **perimeter of isosceles \(\triangle ABC\)** = \(a + a + b = 2a + b\)

Example

**Find perimeter of \(\triangle ABC\) with length of its sides as given below:**

AB=2cm, BC=4cm, CA=6cm

So, perimeter of \(\triangle ABC\) = \(AB + BC + CA\)

= \(2 + 4 + 6\)

= \(12\)cm

Area is the total amount of space occupied by any closed figure.

Area is measured in square units.

The few examples of units of area of triangle are \(meter^2\) or (\(m^2\)) read as meter square, \(centimeter^2\) (\(cm^2\)) read as centimeter square etc..

Area of any triangle = \(\frac{1}{2} \times base \times height\)

Let’s understand it from following \(\triangle ABC\)

Area of triangle \(\triangle ABC\)

where, BC is base, length of BC = b

and AO is height, length of AO = h

Therefore, area of \(\triangle ABC\) = \(\frac{1}{2} \times b \times h\)

Area of equilateral triangle \(\triangle ABC\)

In above equilateral \(\triangle ABC\), where all sides of triangle are equal in length, its area is calculated as:

area of \(\triangle ABC\) = \(\frac{\sqrt 3}{4} \times a^2\)

**Let’s see how it is calculated?**

Here, in right angle \(\triangle AOC\)

\((a)^2 = (h)^2 + (\frac{a}{2})^2\) by Pythagoras theorem

\(a^2 = h^2 + \frac{a^2}{4}\)

\(a^2 – \frac{a^2}{4} = h^2\)

\(\frac{4a^2 – a^2}{4} = h^2\)

\(\frac{3a^2}{4} = h^2\)

\(\frac{\sqrt 3}{2}a = h\)

Area of \(\triangle = \frac{1}{2} \times a \times \frac{\sqrt 3}{2}a\)

Area of \(\triangle = \frac{\sqrt 3}{4}a^2\)

Area of isosceles triangle \(\triangle ABC\)

area of isosceles \(\triangle ABC = \frac{b\sqrt{4a^2-b^2}}{4}\)

**Let’s see how it is calculated?**

Here, in right angle \(\triangle AOC\)

\((a)^2 = (h)^2 + (\frac{b}{2})^2\) by Pythagoras theorem

\(a^2 = h^2 + \frac{b^2}{4}\)

\(a^2 – \frac{b^2}{4} = h^2\)

\(\frac{4a^2 – b^2}{4} = h^2\)

\(\sqrt \frac{4a^2 – b^2}{4} = h\)

\(\frac{\sqrt{4a^2-b^2}}{2} = h\)

we know, area of \(\triangle = \frac{1}{2} \times b \times h\)

\(\therefore \) area of \(\triangle ABC = \frac{1}{2} \times b \frac{\sqrt{4a^2-b^2}}{2}\)

or area of \(\triangle ABC = \frac{b\sqrt{4a^2-b^2}}{4}\)