Types of Triangles Based on Sides: Definitions, Solved Examples

Types of Triangles Based on Sides: A triangle is the simplest form of a polygon. A triangle is a \(2 – \)dimensional shape containing \(3\) sides, \(3\) angles, and \(3\) vertices, but they come in many different shapes and sizes. The sum of all the interior angles of a triangle adds up to \({180^ \circ }.\) Triangles can be divided into six types based on the properties of their angles and their sides.

The understanding of different aspects of a triangle is the backbone of understanding all the other polygons. In this article, we will cover and understand the types of triangles based on sides.

Types of Triangles Based on Sides

Let us have a detailed discussion about the classification of triangles based on their sides one by one.

Equilateral Triangle

A triangle with all the sides equal is called an equilateral triangle.

In the above-given figure, \(\Delta XYZ\) is an equilateral triangle as:
side \(XY\; = \) side \(YZ\; = \) side \(XZ\)
In an equilateral triangle, all the angles are equal, i.e.,  
\(\angle XYZ = \angle YXZ = \angle XZY\)

Since the sum of all the three interior angles of every triangle is \({{{180}^ \circ }},\) therefore each interior angle of an equilateral triangle \( = \;\frac{{{{180}^ \circ }}}{3}\; = \;{60^ \circ }.\)

Area of an Equilateral Triangle

Heron’s formula is used to deduce the formula used to calculate the area of an equilateral triangle.

All the sides are equal in an equilateral triangle. So, in this case, \(a = b = c.\)

So, \(s = \frac{{a + b + c}}{2} = \frac{{a + a + a}}{2} = \frac{{3a}}{2}\)

So, the \({\rm{area}} = \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)}  = \sqrt {\frac{{3a}}{2} \times \left( {\frac{{3a}}{2} – a} \right) \times \left( {\frac{{3a}}{2} – a} \right) \times \left( {\frac{{3a}}{2} – a} \right)} \)

\( = \sqrt {\frac{{3a}}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}  = \sqrt {\frac{{3{a^4}}}{{16}}}  = \frac{{\sqrt 3 }}{4}{a^2},\;\) where \(a\) is the length of the side of the equilateral triangle. Hence, the area of an equilateral triangle \( = \frac{{\sqrt 3 }}{4} \times {a^2} = \frac{{\sqrt 3 }}{4} \times {\left( {{\rm{side}}} \right)^2}.\)

Perimeter of an Equilateral Triangle

The perimeter of a polygon is the sum of the lengths of its sides.

\(\Delta ABC\) is the equilateral triangle with sides \(AB = BC = AC = a.\)
So, the perimeter of \(\Delta ABC = \) the length of the sides \( = AB + BC + CA = a + a + a = 3a\)
\({\rm{ = 3 \times side}}{\rm{.}}\)

Isosceles Triangle

A triangle with at least two sides equal is called an isosceles triangle.

In the above figure, \(\Delta PQR\) is an isosceles triangle, with \(PQ = PR.\) \(\angle Q = \angle R,\) and they are called base angles.

Area of an Isosceles Triangle

Let the two equal sides of an isosceles triangle \(ABC\) are \(AB = AC = a,\) and the length of the base be \(BC = b.\)

Draw \(AD \bot BC.\) So, \(D\) bisects \(AB.\)

Hence, \(BD = \frac{b}{2}.\)

Applying Pythagoras theorem on \(\Delta ABD,\) it can be written that \(A{D^2} + B{D^2} = A{B^2}\)

\( \Rightarrow A{D^2} + {\left( {\frac{b}{2}} \right)^2} = {\left( a \right)^2}\)

\( \Rightarrow A{D^2} = {a^2} – \frac{{{b^2}}}{4}\) \( \Rightarrow AD = \sqrt {{a^2} – \frac{{{b^2}}}{4}} \)

Hence, the area of the isosceles triangle \( = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times BC \times AD\) \( = \frac{1}{2} \times b \times \sqrt {{a^2} – \frac{{{b^2}}}{4}} \)

Perimeter of an Isosceles Triangle

\(\Delta ABC\) is the isosceles triangle with sides \(AB = AC = a\) and \(BC = b.\) So, the perimeter of \(\Delta ABC = \) the length of the sides \( = AB + BC + CA = a + b + a = 2a + b.\)

Scalene Triangle

If all the sides of a triangle are unequal, i.e., if the sides are of different lengths, the triangle is called a scalene triangle.

In a scalene triangle, all the angles are of different sizes, i.e., in \(\Delta ABC,\;\angle A\; \ne \;\angle B \ne \;\angle C\)

Area of a Scalene Triangle

The area of a scalene triangle can be calculated if the measure of all three sides is known or given.

Thus, the area can be calculated as,

\({\rm{Area}} = \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)} \)

where, \(a,\;b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the triangle, given by \(s = \frac{{a + b + c}}{2}.\)

The area of a scalene triangle can also be calculated when the length of any side and the corresponding height is known or given.

For the above figure, the area of the triangle \( = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times BC \times AD = \frac{1}{2} \times b \times h\)

Perimeter of a Scalene Triangle

\(ABC\) is the scalene triangle with sides \(AB = c,\;BC = a\) and \(AC = b.\) So, the perimeter of \(\Delta ABC = \) the length of the sides \( = AB + BC + CA = c + a + b = a + b + c.\)

Types of Triangles Based on Angles

Let us have a detailed discussion about the classification of triangles based on their angles.

Acute-angled Triangle

If each angle of a triangle is acute (less than \({90^ \circ }\)), it is called an acute-angled triangle.

Right-angled Triangle

If one of the angles of a triangle is a right angle, i.e., \({90^ \circ },\) it is called a right-angled triangle. In the right-angled triangle, the side opposite to the right angle is called the hypotenuse. The hypotenuse is the largest side of a right-angled triangle.

Obtuse-angled Triangle

If any angle of a triangle is obtuse (more than \({90^ \circ }\)), the triangle is called an obtuse-angled triangle.

Properties Related to the Sides of a Triangle

1. In any triangle, the sum of any two sides of a triangle is greater than the third side.
2. In any triangle, the longest side is opposite to the biggest angle.
3. The difference between the two sides of a triangle is always less than the third side in any triangle.

Solved Examples on Types of Triangles Based on Sides

Q.1. Classify the following triangle.

Ans: We can distinguish between equilateral triangle, isosceles triangle and scalene triangle based on the measurement of their sides. The given figure is of a scalene triangle because all the sides and angles have different measurements.

Q.2. Classify the triangle given below.

Ans: In the given figure, two angles are of equal measurement. Hence, the opposite sides to these angles will also be equal. Thus, this is an isosceles triangle.

Q.3. Find the area of a triangle whose lengths of the sides are \({\rm{3\;cm,\;4\;cm}}\) and \({\rm{5\;cm}}{\rm{.}}\)
Ans: Given the sides of triangle \(a = 3\;{\rm{cm}},\;b = 4\;{\rm{cm}}\) and \(c = 5\,{\rm{cm}}.\) As the three sides are of different measures, this is a scalene triangle. So, the area of this triangle can be found by Heron’s formula.

\({\rm{Area}} = \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)} .\)

where, \(a,b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the triangle, given by \(s = \frac{{a + b + c}}{2}.\)

So, \(s = \frac{{a + b + c}}{2} = \frac{{3 + 4 + 5}}{2} = \frac{{12}}{2} = 6{\rm{\;cm}}\)

Hence, the area of the given triangle \( = \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)}  = \sqrt {6\left( {6 – 3} \right)\left( {6 – 4} \right)\left( {6 – 5} \right)} \)

\( = \sqrt {6 \times 3 \times 2 \times 1}  = \sqrt {36}  = 6\;\)
Hence, the area of the given triangle is \({\rm{6\;c}}{{\rm{m}}^{\rm{2}}}{\rm{.}}\)

Q.4. The equal sides of an isosceles triangle are \({\rm{10\;cm}}\) each, and the base is \({\rm{4\;cm}}{\rm{.}}\) Find the area of the triangle.
Ans: Here, the measure of equal sides, \(a = 10\;{\rm{cm}}\)
Base, \(b = 4{\rm{\;cm}}\)
Hence, the area of the isosceles triangle \( = \frac{1}{2} \times b \times \sqrt {{a^2} – \frac{{{b^2}}}{4}}  = \frac{1}{2} \times 4 \times \sqrt {{{10}^2} – \frac{{{4^2}}}{4}} \)
\( = 2 \times \sqrt {100 – \frac{{16}}{4}}  = 2 \times \sqrt {100 – 4}  = 2 \times \sqrt {96}  = 2 \times \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 3}  = 2 \times 4\sqrt 6 \;\)
\( = \;8\sqrt 6 \;\)
Hence, the area of the given triangle is \(8\sqrt 6 \;{\rm{\;c}}{{\rm{m}}^{\rm{2}}}.\)

Q.5. Find the area of an equilateral triangle of side \(10\;{\rm{cm}}{\rm{.}}\)
Ans: Here, \(a = 10\;{\rm{cm}}\) Hence, the area of an equilateral triangle with a side of \(10\;{\rm{cm}}\) \( = \frac{{\sqrt 3 }}{4} \times {a^2} = \frac{{\sqrt 3 }}{4} \times {\left( {10} \right)^2} = \frac{{\sqrt 3 }}{4} \times 100 = 25\sqrt 3 \;{\rm{c}}{{\rm{m}}^{\rm{2}}}.\)

Summary

A triangle can be defined as the simplest form of a polygon. Triangles can be classified into various types depending on triangles based on angles and based on sides. The triangles based on angles are classified into three types namely, acute-angled triangle, right-angled triangle, and obtuse-angled triangle. The triangles based on sides can be classified into three types namely, isosceles triangle, equilateral triangle and scalene triangle.

Furthermore, it is important to note that in any triangle, the sum of any two sides of a triangle is greater than the right side. The longest side is opposite to the biggest angle in any triangle.

FAQs on Types of Triangles Based on Sides

Q.1. Explain types of triangles based on angles?
Ans: Based on angles, a triangle can be classified as:
1. Acute-angled triangle: If each angle of a triangle is acute (less than \({90^ \circ }\)), it is called an acute-angled triangle.
2. Right-angled triangle: If one of the angles of a triangle is a right angle, i.e., \({90^ \circ },\) it is called a right-angled triangle.
3. Obtuse-angled triangle: If the angles of a triangle are obtuse (more than \({90^ \circ }\)), the triangle is called an obtuse-angled triangle.

Q.2. What is a triangle and define its angle sum property?
Ans: A triangle is the smallest polygon with three sides, three vertices and three angles. The angle sum property of a triangle states that the sum of three internal angles is \({180^ \circ }.\)

Q.3. Explain types of triangles based on sides?
Ans: Based on sides, a triangle can be classified as:
1. Isosceles triangle:  A triangle with at least two sides equal
2. Equilateral triangle: A triangle with all the sides equal
3. Scalene triangle: If the three sides of a triangle are unequal, i.e., if the sides are of different lengths

Q.4. What are the types of a triangle?
Ans: The types of triangles are based on parameters. Below we have listed the types of triangles according to the various parameters:
1. Based on sides
a. Scalene Triangle
b. Isosceles Triangle
c. Equilateral Triangle
2. Based on angles
a. Acute Angle Triangle
b. Right Angle Triangle
c. Obtuse Angle Triangle

Q.5. How do you know if the given sides can form a triangle?
Ans: In any triangle, the sum of any two sides of a triangle is greater than the third side. Also, the difference between the two sides of a triangle is always less than the third side in any triangle. By verifying this, we can find whether the given sides form a triangle or not.

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