Triangles: Learn the Definition, Various Types, Properties and Formulas
Depanjan Chowdhury
| Updated: Apr 20, 2021 20:58 IST
A triangle is a polygon with three sides, three vertices and three edges, also any three non-collinear points always create a triangle. Similarly, any four non-collinear points build a quadrilateral in mathematics. This is an important part of geometry, as topics like Pythagoras theorem and related trigonometric identities are derived using these triangles and related properties. With this article, you will learn about the various types and parts of triangles followed by the various properties, formulas and related terms like centroid, incenter, circumcentre, etc with solved examples and more.
What is a Triangle?
Consider a triangle with 3 vertices says P, Q, and R are represented as △PQR (where △ represents the symbol for a triangle). In a particular triangle, the total sum of the internal angle of a triangle is equal to 180 degrees. This is also known as the angle sum property of a triangle. A triangle is divided into different types based on the angles and sides. You will learn about all these types in the coming headers.
Parts of Triangle
A closed figure with three angles when three line segments are joined end to end forms a triangle shape. Thus, we can assume that a triangle is a polygon, which has 3 sides, 3 angles, and 3 vertices as shown in the figure.
For the above figure:
- The three angles of a triangle are, ∠PQR, ∠QRP, and ∠RPQ respectively.
- The three sides of a triangle are side PQ, side QR, and side RP.
- The three vertices of a triangle are P, Q, and R.
Types of Triangles
Triangles can be classified; on the ground of angle and on the basis of the length of their sides. Below is the tabular representation of various types of triangles in mathematics based on the classification with detailed images.
| Type | Definition | Representation by Image |
Types of Triangles based on the Sides | ||
| Scalene Triangle | A triangle with all three sides of varying lengths is a scalene triangle. As all the three sides are of distinct lengths, the three angles will also be different. |  |
| Isosceles Triangle | A triangle with two sides of equal length and the third side of a varying length is an isosceles one. The angles that are opposite to the equivalent sides are also equal in measure. |  |
| Equilateral Triangle | A triangle with all 3 sides of an identical length is equilateral. As all three sides are of equal length, therefore, all the three angles will also be identical. |  |
Types of Triangles based on the Angles | ||
| Acute Angled Triangle | A triangle with all three angles smaller than 90° is an acute angle triangle. Therefore, all the angles of an acute angle triangle are termed acute angles. |  |
| Right Angled Triangle | A triangle with one angle that exactly measures 90° is called a right-angle triangle. The remaining two angles of a right-angle triangle are acute. |  |
| Obtuse Angled Triangle | A triangle with one of the three angles more than 90° is obtuse. |  |
Properties of a Triangle
Some of the important properties of a triangle are as follows:
- A triangle has three sides, angles, and vertices respectively. The total length of any two sides of a triangle is larger than the measure of the third side. This is also known as the triangle inequality property.
- The total of all internal angles of a triangle is always equal to 180 degrees. This is also known as the angle sum property of a triangle.
- Also, the difference between the two sides of a triangle is lesser than the length of the third side.
- The exterior angle property of a triangle states that any exterior angle of the triangle is equivalent to the sum of its interior opposite angles.
- Two triangles are supposed to be similar triangles if the corresponding angles of both triangles are congruent and the measures of their sides are proportional.
- The Congruence Property states that two triangles are congruent if all their corresponding sides and angles are identical.
The total of all exterior angles of any triangle is equivalent to 360°.
Triangle Formula
Be it a school exam or competitive exams like SSC or RRB, questions from trigonometry/geometry/ polygons are always asked. Therefore, to have a good command of the subject, one must have excellent knowledge of the formula. So, let us now discuss formulas.
Perimeter of Triangle
The perimeter of a triangle is simply the entire length of the outer boundary of the triangle that is;
∴ Perimeter = Total sum of all Sides.
The perimeter of a triangle=P(perimeter)=a+b+c units
Where a,b and c are the sides of the triangle.
Area of Triangle Formula
The area of a triangle is the region occupied by the triangle, the area varies from triangle to triangle. The area(measured in square units) can be calculated with the knowledge of the base and the height.
\(\text{Area of a Triangle }\left(A\right)=\frac{1}{2}\times b\left(base\right)\times h\left(height\right)\)
Heron’s Formula
If the height of a triangle in maths is not provided, then the above formula cannot be used to find the area of a triangle. Here comes the concept of Heron’s formula.
Heron’s formula can be used to calculate the area of a triangle if the length of all the sides is given.
\(A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\)
Where s is the semi-perimeter of the triangle
\(s=\frac{(a+b+c)}{2}\)
Check out this article on circles here.
Centroid of a Triangle
The term centroid is defined as the centre point of the object. The point at which the three medians of the triangle intersect or touch each other is recognised as the centroid of a triangle.
The point of intersection of medians of a triangle is called the centroid of the triangle in maths. If the coordinates of the vertices of triangle Δ ABC are: A \((x_1, y_1)\), B \((x_2, y_2)\), and C \((x_3, y_3)\). Then the coordinates of the centroid G is given by:
⇒\(\left(\frac{x_1+x_2+x_3}{3},\ \frac{y_1+y_2+y_3}{3}\right)\)
Here \(x_1,\ x_2\ \text{and}\ x_3\) are the x-coordinates of the vertices of the triangle.
and \( y_1,\ y_2\ \text{and}\ y_3\) are the y-coordinates of the vertices of the triangle.
Incenter of a Triangle
The point of intersection of angle bisectors of a triangle is called the incenter of the triangle in maths/ the center of the circle which touches the sides of a triangle internally is called the incenter of the triangle as shown in the figure.
Check out this article on the Binomial Theorem.
Circumcentre of a Triangle
The point of intersection of the perpendicular bisectors of sides of a triangle is called the circumcenter of the triangle in maths/ the center of the circle which passes through the vertices of a triangle is called the circumcenter of the triangle as shown in the figure.
Altitude of a Triangle
A perpendicular line drawn from the vertices of the triangle to the opposite side of the triangle is known as the altitudes of the triangle in maths.
Orthocentre of a Triangle
The orthocenter is defined as the point of intersection of the altitudes of the triangle in maths.
Solved Examples on Triangles
At this point of our discussion, you can answer questions like what is the triangle definition, how many sides a triangle have, and how many types of triangle are there. Let us step towards some solved examples to understand these elements.
Solved Example 1: The measure of two angles of a triangle is \(65^{\circ}\) and\(75^{\circ}\). What will be the measure of the third angle?
Solution: The measures of two angles of a triangle are \(65^{\circ}\text{ and }75^{\circ}\).
Sum of the measures of two angles\(=65^{\circ}+75^{\circ}=140^{\circ}\).
Sum of all three angles of a triangle by angle sum property =\(180^{\circ}\).
Hence, the measure of the third angle \(=180^{\circ}-140^{\circ}=40^{\circ}\).
Check out this article on Family of Lines.
Solved Example 2: If the measure of the sides of a triangle is 3cm,4cm and 5cm(where 4cm is the base of the triangle) with the altitude of the triangle is 3.2 cm, then obtain the area of the triangle?
Solution:
Given:
Base=4cm.
Height=3.2 cm.
Area of a Triangle(A)=\(\frac{1}{2}\times b\left(\text{base}\right)\times h\left(\text{height}\right)\)
\(A=\left(\frac{1}{2}\right)\times4\times3.2\)
\(A=6.4\ cm^2\)
Know more about Geometric Shapes here.
Solved Example 3: Find the area of a triangle by Heron’s formula where the sides of the triangles are 4,5 and 7 cm respectively.
Solution: By Heron’s formula:
Semiperimeter (s) = \(\frac{\left(a+b+c\right)}{2}\)
\(s=\frac{\left(4+5+7\right)}{2}=\frac{16}{2}=8\)
Now the area of a triangle = \(\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\)
\(=\sqrt{8\left(8-4\right)\left(8-5\right)\left(8-7\right)}\)
\(=\sqrt{8\times4\times3\times1}\)
\(=\sqrt{2\times2\times2\times2\times2\times3\times1}\)
\(=\sqrt{2^2\times2^2\times2\times3\times1}\)
\(=4\sqrt{6}\ cm^2\)
We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.
| If you are checking Triangles article, also check related maths articles: | |
| Corresponding Angles | Similar Triangles |
| Area of Similar Triangles | Similar Figures |
| Congruent Figures | Parallelogram |
Triangles FAQs
Q.1 What is a triangle?
Ans.1 A triangle is a type of polygon with three sides, three edges and three vertices.
Q.2 What is the important property regarding the sides of a triangle?
Ans.2 The total lengths of any two sides of a triangle are greater than the magnitude of the third side. In the same way, the difference between the lengths of any two sides of a triangle is smaller than the length of the third side.
Q.3 How are triangles classified based on the angle?
Ans.3 The triangles are classified as an obtuse-angled triangle (one angles more than 90°), a right-angled triangle(one of the angles is equal to 90°), and an acute-angled triangle(all angles less than 90°) based on the angle.
Q.4 What are the values of isosceles triangle angles?
Ans.4 In an isosceles triangle, the angles opposite the equivalent sides are equal in measure.
Q.5 What are the three types of triangles based on sides?
Ans.5 Types of triangles based on the sides: are; scalene triangle, isosceles triangle and equilateral triangle.
Q.6 What is a scalene triangle?
Ans.6 A triangle with all three sides of varying lengths is a scalene triangle. As all the three sides are of distinct lengths, the three angles will also be different.
