Table of Contents
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Introduction to Triangle
A closed figure made with 3 line segments forms the shape of a triangle.
Triangles are an important part of our lives.
You can observe the shape of a triangle in signboards, your favourite sandwiches or birthday banners.
What is a Triangle?
A triangle is a closed figure or shape with 3 sides, 3 angles and 3 vertices.
A triangle with vertices P, Q, and R is denoted as △PQR.
The elements of a triangle are its 3 angles and its 3 sides.
Let's look at triangle PQR.
The elements of this triangle are:
 \(\angle PQR\)
 \(\angle QRP\)
 \(\angle RPQ\)
 PQ
 QR
 RP
You can explore the relation between the elements of a triangle by dragging the vertices of the triangle shown below.
Different Types of Triangles
Triangles can be classified based on their angles or the length of their sides.
Classification of Triangles by Angles
Type of Triangle  Example  Explanation 

Rightangled triangle 
A right triangle has one of its vertex angles as 90°. In the above triangle, \(\angle \text{ACB}\) = 90°. 

Acuteangled triangle 
An acute triangle has all its internal angles as acute( i.e.. less than 90°). 

Obtuseangled triangle 
An obtuse triangle has one of its internal angles as obtuse ( i.e.. greater than 90°). In the given triangle, \(\angle \text{DFE}\)= 120\(^\circ\) 
Classification of Triangles by Sides
Type of Triangle  Example  Explanation 

Equilateral triangle 
If all the three sides of a triangle are equal, it is called an equilateral triangle. In the given triangle, all three sides are equal in length. AB = BC = AC


Isosceles triangle 
If any two sides of a triangle are equal, it is called an isosceles triangle. In the given triangle, two sides are equal. RS = RT


Scalene triangle 
If a triangle has three unequal sides, it is called a scalene triangle. In the given triangle, all three sides are of unequal lengths. XZ ≠ XZ ≠ YZ

Fun Facts on Triangles
 A triangle can always be divided into two right triangles, irrespective of its orientation.
 Triangles are polygons with the least number of sides. There are no polygons with less than 3 sides.
 Many towers, buildings and monuments are constructed in the shape of triangles to give them more stability and strength.
Here are few examples for you:
1. Eiffel Tower
The worldfamous Eiffel Tower is triangular in shape and has around 186 triangles in its structure
2. Egyptian Pyramids
They have triangular faces
3. Bridges
They use triangular frames for stability
Properties of a Triangle
Some of the important properties of a triangle are listed below.
Property 1  Angle Sum Property
The Angle Sum Property states that the sum of the three interior angles of a triangle is 180°
In the given triangle,
\(\angle \text{D} + \angle \text{E} + \angle \text{F} = 180^\circ \)
Property 2  Triangle Inequality Property
The Triangle Inequality Property sum of the length of the two sides of a triangle is greater than the third side.
in \(\Delta PQR \)
 p + q > r ( 6 + 4 > 3)
 r + p > q (3 + 4 > 6)
 r + q > p ( 6 + 3 > 4)
Property 3  Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.
\(\begin{align}\text{Hypotenuse}^2 &\!=\!\text{Base}^2 \!+\!\text{Altitude}^2 \end{align}\) 
Property 4  Side opposite the greater angle is the longest side
To understand the side opposite the greater angle is the longest side property, let's consider the belowgiven triangle.
In this triangle, \(\angle \text{A}\) is the greatest angle.
Thus, side BC is the longest side.
Property 5  Exterior Angle Property
The Exterior Angle Property of a triangle is always equal to the sum of the interior opposite angles.
In the given triangle, Exterior angle E1 = \(\angle \text{ABC} + \angle \text{BCA}\)
A triangle has 3 exterior angles and these exterior angles add up to 360° for any polygon.
Property 6  Congruence Property
In the Congruence Property, two triangles are said to be congruent if all their corresponding sides and angles are congruent.
All the corresponding angles and sides are equal,
 \(\angle\text{ ABC} = \angle\text{PQR}\)
 \(\angle\text{ BAC} = \angle\text{QPR}\)
 \(\angle\text{ BCA} = \angle\text{QRP}\)
 AB = PQ
 AC = PR
 BC = QR
Formulas of a Triangle
The formulas of a triangle help us determine its area and perimeter.
Perimeter of a Triangle
Consider △XYZ.
The length of the sides of the triangle are represented as ‘a’, ‘b’ and ‘c’ respectively.
\(\text{Perimeter of Triangle} = a + b + c\) 
Area of a Triangle
In the given triangle \(ABC\),
\(\begin{align}\text{Area of }\Delta ABC = \frac{1}{2} h\times \text{BC}\end{align}\)
\(\text{BC}\) is the base and \(h\) is the height of the triangle.
\(\begin{align}\frac{1}{2} \times \text{Base} \times \text{Height}\end{align}\) 
 A triangle cannot have more than one right angle.
 Each angle of a triangle cannot be less than 60°.
 The angles opposite to the equal sides of an isosceles triangle are equal.
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Solved Examples
Example 1 
The perimeter of a triangular garden is given as 13 m.
If two of its sides measure 3 m and 4 m respectively, what is the measure of the third side?
Solution:
We know that the perimeter of a triangle is the sum of all three sides
\(\implies\) 13 = 3 + 4 + unknown side
Therefore, the unknown side is given by:
\(\begin{align} &=13 \!\! (3 \!+\! 4) \\ &= 6 \;m \end{align}\)
\(\therefore\)The third side of the park measures 6 m 
Example 2 
Tina is building a triangular, wooden birdhouse as shown.
If two of the angles measure 46° and 62°, what is the measure of the third angle?
Solution:
We know that the sum of the angles of a triangle add up to 180°.
Therefore, the unknown angle is:
\(\begin{align} &=180^\circ \!\! (46^\circ \!+\! 62^\circ) \\ &= 72^\circ\end{align}\)
\(\therefore\)The third angle measures 72° 
Example 3 
In a right triangle, two of the sides measure 3 cm and 4 cm.
What is the measure of the hypotenuse?
Solution:
By Pythagoras theorem, we know that, hypotenuse^{2 }= base^{2} + height^{2}
Substituting the values, we get:
\(\begin{align}
\text{hypoteneuse}^2 &=3^2 +4^2 \\
&= 9 + 16\\
&= 25\: \text{cm} \\
\implies \text{hypoteneuse} &= 5 \:\text{cm}
\end{align}\)
\(\therefore\) Hypotenuse = 5 cm 
Example 4 
One of the acute angles of a rightangled triangle is 45°
Find the other triangle.
Identify the type of triangle thus formed.
Solution:
Given, angle_{1} = 90° and angle_{2 }= 45°
We know that the sum of the angles of a triangle add up to 180°.
Therefore,
\(\begin{align} \text{angle}_3 &=180^\circ \!\! (90^\circ \!+\! 45^\circ) \\ &= 45^\circ\end{align}\)
Since two angles measure the same, it is an isosceles triangle.
\(\therefore\) △ABC is an isosceles triangle 
Example 5 
The sides of a triangular park are in the ratio 3: 4: 5 and its perimeter is 1080 m.
Find the area of the triangular park.
Solution:
Given:
 Perimeter = 1080 m
 Sides of the triangle are in the ratio 3:4:5.
Therefore, the sides of triangle are
\( \begin{align}
\text{side}_1 &= \frac{3}{12}\times 1080\\\ &= 270 \text{m}\\\\
\text{side}_2 &= \frac{4}{12}\times 1080\\\ &= 360\text{m} \\\\
\text{side}_3 &= \frac{5}{12}\times 1080\\\ &= 450\text{m}
\end{align} \)
\(\begin{align} \text{Area of the park} &= \frac{1}{2} \times 270 \times 360 \\
&= 48600\: \text{m}^2\end{align} \)
\(\therefore\)Area of the park = 48,600 m^{2} 
 A triangle is a 3sided closed shape.
 The perimeter of a triangle is the sum of all three sides.
 The area of a triangle is half the product of its height and base.
 Herons formula and Pythagoras theorem are important formulas related to triangles.
 The sum of the angles of a triangle add up to 180°.
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Practice Questions
Here are few questions for you to practice.
Select/Type your answer and click the "Check Answer" button to see the result.
Area of a triangle worksheets are available at the end of the page for you to try and solve!
1.  Anu said a triangle has at least one set of parallel lines. Is she right? Why? 
2.  Can a triangle be formed with dimensions 17cm, 9 cm and 30cm? 
Important Topics
Given below are the list of topics that are closely connected to triangles. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
Maths Olympiad Sample Papers
IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.
You can download the FREE gradewise sample papers from below:
 IMO Sample Paper Class 1
 IMO Sample Paper Class 2
 IMO Sample Paper Class 3
 IMO Sample Paper Class 4
 IMO Sample Paper Class 5
 IMO Sample Paper Class 6
 IMO Sample Paper Class 7
 IMO Sample Paper Class 8
 IMO Sample Paper Class 9
 IMO Sample Paper Class 10
To know more about the Maths Olympiad you can click here
Frequently Asked Questions(FAQs)
1. What is a triangle?
A threesided closed shape made with 3 line segments is called a triangle.
2. What are the types of triangles?
 Types of triangles based on sides:
 Scalene
 Isosceles
 Equilateral
 Types of triangles based on angles:
 Acute triangle
 Obtuse triangle
 Right triangle
3. How many properties does a triangle have?
The 6 important properties of triangles are listed in section Properties of a Triangle.