## Triangle Definition

In the geometry of mathematics, we come across different types of shapes - some two dimensional and some three dimensional. Shapes that can be defined with the help of two parameters are called two-dimensional shapes and those which can be defined with three parameters form three-dimensional shapes. Two-dimensional shapes include squares, rectangles, triangles, etc. A polygon is such a two-dimensional structure having straight sides. A triangle consists of three line segments placed at a particular angle for any pair of the three line segments. A triangle has three angles whose sum is constant which is 180°.

Let a triangle be △PQR.

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The sum of the angles of △PQR is 180°.

Hence, ∠P + ∠Q + ∠R = 180°.

### Types of Triangle

Based on side lengths of a triangle, triangles can be classified as-

Equilateral Triangles

Triangles in which all the sides of the triangle are equal in length.

Isosceles Triangles

Triangles in which any two sides of the triangle are equal resulting in equal base angles.

Scalene Triangle

Triangles in which none of the sides has an equal length or all the sides of the triangle are different in lengths.

Based on the interior angles of a triangle, triangles can be classified as -

Acute Angled Triangle

Triangles in which all the angles are less than 90°.

Right Angled Triangle

Triangles in which any one of the angles is exactly 90°.

Obtuse Angled Triangle

Triangle in which any one of the angles is more than 90°.

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### Triangle Formula

If ∠A, ∠B, and ∠C are the interior angles of a triangle, ∠A + ∠B + ∠C = 180°.

If b denotes the base of a triangle and h denotes the altitude of the triangle, the area of the triangle is given 12 х base х height square units.

If a,b and c are the sides of a triangle, the area of the triangle can be given by Heron’s formula = \[\sqrt{s(s - a)(s - b)(s - c)}\]

(where s stands for semi perimeter which is 12(a+b+c)

If side lengths of a triangle are denoted by a, b and c, then the perimeter of the triangle is given by (a+b+c).

In a right-angled triangle, if p denotes perpendicular, b denotes base, and h denotes hypotenuse, h2= b2+ p2. This is known as Pythagoras’ Theorem.

### Solved Examples

1. Calculate the area of a right-angled triangle in which the measurement of the hypotenuse is 5cm and that of the base is 3cm.

Ans: Given, base = 3cm, hypotenuse = 5cm.

We are given a right-angled triangle whose hypotenuse and base are known.

To find out the area of a triangle, we either need to know the values of base and height or the lengths of all the sides of the triangle. To find out the perpendicular of the triangle, we may use the Pythagoras theorem. The perpendicular of a right-angled triangle corresponds to the altitude of a triangle.

Let altitude/perpendicular be p cm.

For a right - angled triangle,

By Pythagoras Theorem,

Perpendicular2 = hypotenuse2 - base2

Perpendicular2 = 52 - 32

= 25 - 9

Perpendicular2 = 16

perpendicular = 4

\[\text{Area of the Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 4 = 6\]

Hence, the area of the triangle is 6 cm.

**2. The perpendicular of a right-angled triangle is 8 units. The area of the triangle is given to be 24 square units. Find the base, hypotenuse, and perimeter of the triangle.**

Ans: Given, perpendicular = 8 units, area = 24 square units

We can find out the base of the triangle from the area formula and hypotenuse by Pythagoras Theorem consequently.

We know,

\[Area = \frac{1}{2} \times base \times height\]

\[base = \frac{2 \times Area}{height}\]

So,

\[= \frac{2 \times 24}{8} = 6\]

Now we know the base and perpendicular of the triangle.

By Pythagoras theorem,

Perpendicular2 = hypotenuse2 - base2

Perpendicular2 = 82 - 62

= 64 - 36

Perpendicular2 = 100

perpendicular = 10

Now we know all the three sides of the triangle.

Perimeter = hypotenuse + base + perpendicular

=> Perimeter = 10 + 6 + 8 units

=> Perimeter = 24 units.

Hence, the base is 6 units, the hypotenuse is 10 units and the perimeter of the triangle is 24 units.

### Points To Remember

A triangle is a three shaped polygon with three sides, and the two sides joined on the ends is known as the vertex.

An angle is formed between the two sides.

Triangle is a closed two-dimensional shape and all the sides are made of straight lines.

The point where two straight lines join is known as the vertex. Therefore, the triangle has three vertices with each vertex forming an angle.

The properties of triangles distinguish them from other shapes.

Triangles have three angles as well as three sides.

180 degrees is the sum total of the angles of the triangle.

360 degrees is the sum of exterior angles of a triangle.

The interior and exterior angles have a supplementary sum.

The perimeter of a triangle is the length of the boundary of the triangle on the outside.

The region occupied by the triangle in a two-dimensional space is known as Area.

Isosceles, equilateral and Scalene triangles are the kinds of triangles that are different from one another on the basis of the length of their sides.

In the scalene triangle, all three sides are different in length.

In an Isosceles triangle, any two sides are equal in length.

In an Equilateral triangle, all three sides are equal in length.

In an acute triangle, all the angles present in it are less than 90 degrees.

In an obtuse triangle, any one of the angles is greater than 90 degrees.

In a right-angled triangle, one angle is equal to 90 degrees.

### Tips To Master Triangles

**Remember the Rules**

Students must pay closer attention to the rules of triangles in order to solve the sums more effectively and within less time. Calculating the angles and sides can be time-consuming so a student must be well versed with the rules and criteria.

**Practice Religiously**

Maths is a subject that requires constant practice and so does this chapter. In order to solve the sums like an expert, a student must practice the sums regularly to stay in touch with the concepts and rules of the triangle.

**Solve Questions from RD Sharma Book.**

A student should solve the questions from the textbook in order to prepare for the exam. Practising sums of triangles from RD Sharma can give an idea of possible questions to the students.

**Practice Solved Examples**

The solved examples establish the fundamental knowledge and idea about triangles for the students. It can be helpful in understanding the basic concepts of the chapter including the rules and types of triangles.

**Attend Classes on Triangles**

Paying attention to the teacher and the classes can help a student understand the chapter in the best possible way. A student must be willing to listen and learn carefully about triangles in order to solve the sums and calculate the area, sides and perimeter of the triangle.

**Make Notes During Class**

A student must make notes simultaneously during the classes so that the notes can help later during revision sessions. It is very important to note the types and the formulas in order to calculate the sums correctly.

**Stay in Touch with Congruent Triangles with the Help of Sample Papers**

A student must be willing to solve the sample papers after solving the examples and questions from the textbook in order to truly master the chapter. Triangles can have various possible questions for exams so it is always better to be prepared.

**Test Knowledge with Quizzes**

After solving questions from the books, students may test their knowledge through quizzes which will help them clock their time taken for solving one sum. It is a great way to challenge their skills on a lighter note.

**Participate in Online Interactive Classes**

Students can participate in online classes in order to clear their doubts as well as for the revision of this chapter. It will surely help the students understand types of triangles along with their formulas in more efficient ways. It will give them room to make notes and answer the questions to test their skills.

Students can access the online materials such as notes, practice papers, sample questions, solutions to RD Sharma book along with puzzles and quizzes on Congruent Triangles from Vedantu’s official website or mobile application. The solutions can be downloaded in PDF formats as well. If a student is stuck at any point in Congruent Triangles then the student can access the online live classes and doubt clearing sessions in order to progress in the chapter.

## FAQs on Triangle Definition in Maths

**1. Discuss the Various Properties of Triangles.**

There are certain properties of the triangle which has been discussed below-

Triangles are polygons with the minimum number of sides.

Every triangle has three sides, three vertices, and three interior angles.

No matter how a triangle is constructed, the sum of the interior angles of the triangle will always be 180°.

The sum of lengths of any two sides of a triangle will always be greater than the length of its third side.

The difference of lengths between any two sides of a triangle will always be lesser than the length of its third side.

Any triangle can be divided into two right triangles in any given orientation.

The interior angles of an equilateral triangle are equal which is 60°.

**2. What are Some of the Applications of Triangles?**

Triangle is a very important shape and has numerous applications in Mathematics and real life.

Triangles are used in calculus. We use triangles as an elementary part of a different structure which helps in calculating the various properties of the structure. For instance, it is used to evaluate the area of a circle.

A polygon may be broken into simpler triangles and we can find the area of an n-sided polygon.

Triangle is an important shape in the construction of different real-life structures. It is one of the strongest shapes and has numerous applications in the field of construction.

Right-angled triangles combined with trigonometry is used to solve several real-life application based problems involving heights and distances and angle of elevation and angle of depression.