Triangles: Definition, Formula, Types & Properties

Triangle is a geometrical figure with three angles, three sides and three vertices. If ABC is a triangle, the vertices can be represented as A, B, and C. The basic property is that the sum of all internal angles of a triangle equals 180 degrees.  Understanding different aspects of the triangle contribute towards developing the backbone of understanding all other polygons, the entire geometry at large and many other theories and practices of mathematics, physics etc. This article will help you learn about different types of triangles, their properties, and different formulas.

Students studying in the CBSE board can access different study materials offered by Embibe. These study materials include PDF of NCERT books, previous year question papers and solution sets. These solution sets were prepared by a team of experts who are well acquainted with the exam pattern and the marking scheme followed by CBSE board. Each question is explained in detail and helps students to understand the correct approach to be followed to answer the questions correctly.

Triangles: Definition

A triangle is a polygon that has three sides. It can also be defined by a figure bounded or enclosed by three-line segments.

Clearly, a triangle will have three sides and three vertices.

Shapes of Triangle

A triangle is a \(2\)-dimensional closed polygon consisting of three straight sides. Each pair of two consecutive sides forms a vertex and an angle. The shapes of triangles can vary. A few triangle shapes are given below:

Triangle is a closed two-dimensional shape. It is a three-sided polygon. All sides are made of straight lines. The point where two straight lines join is the vertex. Hence, the triangle has three vertices. Each vertex forms an angle.

Triangles: Types

The triangles are classified into two broad types:

a) types of triangles based on the length of sides of the triangle

b) types of triangles based on the internal angles of the triangle

Triangles: Types Based on Length of Sides

Based on the length of the sides, the triangles are classified into three types: scalene triangle, isosceles triangle, and equilateral triangle.

Scalene Triangle: If all the three sides of the triangle are different in length or if none of the sides of the triangle is equal, then the triangle is called a scalene triangle. The triangle given below is a scalene triangle.

In this triangle, all three angles are also different in measure.

Isosceles Triangle: If any two of the three sides of a triangle are equal, then the triangle is called an isosceles triangle.

In the above triangle, the two sides that are equal are indicated. It is an isosceles triangle. In an isosceles triangle, the two angles opposite to the two equal sides are equal in measure.

Equilateral Triangle: If all the three sides of a triangle are the same in length, then the triangle is called an equilateral triangle.

In an equilateral triangle, the measure of all the angles is also the same. It is \({60^{\rm{o}}}.\)

Triangles: Types Based on Angles

Based on the measure of the internal angles, the triangles are classified into three types: acute-angled triangle, right-angled triangle, and obtuse-angled triangle.

Acute Angled Triangle: If all the three angles in a triangle are acute angles, then the triangle is called an acute-angled triangle.

(An angle measuring more than \({{\rm{0}}^{\rm{o}}}\) but, less than \({\rm{9}}{{\rm{0}}^{\rm{o}}}\) is called acute angle)

Right-Angled Triangle: If one angle in a triangle is a right-angle, then the triangle is called a right-angled triangle.

(An angle whose measure is exactly \({\rm{9}}{{\rm{0}}^{\rm{o}}}\) is called a right-angled triangle.). The side opposite to the right angle is called the hypotenuse of the triangle. The other two angles measure less than \({\rm{9}}{{\rm{0}}^{\rm{o}}}\) each.

Obtuse Angled Triangle: If one angle in a triangle is an obtuse-angle, then the triangle is called an obtuse-angled triangle.

(An angle whose measure is more than \({\rm{9}}{{\rm{0}}^{\rm{o}}}\) and less than \({\rm{18}}{{\rm{0}}^{\rm{o}}}\) is called an obtuse angle.). Clearly, the other two angles are acute angles.

Triangles: Properties

A triangle has a few characteristic properties, which have made it different from the other polygons. They are:

1. A triangle has three sides, three vertices and three angles.
2. The sum of all the internal angles in a triangle is \({\rm{18}}{{\rm{0}}^{\rm{o}}}\) always.
3. If all the sides of a triangle are of different lengths, its angles also have different measures such that the greater side is opposite to the greater angle.
4. If all the three angles in a Triangle are different, then the length of the sides are also different. Then, the greater side is at the opposite of the greater angle.
5. In a triangle, if any two sides are equal in their lengths, then the two angles opposite to these equal sides are also equal. The reverse is also true. If any two angles in a triangle have the same measure, then the two sides opposite them are equal.
6. If all the sides in a triangle are equal in measure, all the angles are also equal. And, if all the angles in a triangle have the same measure, then the side lengths are also equal.
7. The exterior angle in a triangle equals the sum of the two interior angles in a triangle that are not adjacent to the exterior angle.

Triangle: Perimeter

The perimeter of a triangle is the total length of boundary of the triangle. The perimeter is obtained by adding the lengths of the three sides of the triangle.

Measuring the Perimeter of a Scalene Triangle

\(ABC\) is the scalene triangle with sides \(AB=c, BC=a\) and \(AC = b.\)

So, the perimeter of \(ABC= \) the length of the sides \( = AB + BC + CA = c + a + b = a + b + c.\)

Measuring the Perimeter of an Isosceles Triangle

\(ABC\) is the isosceles triangle with sides \(AB=AC=a\) and \(BC = b.\)

So, the perimeter of \(ABC=\) the length of the sides \( = AB + BC + CA = a + b + a = 2\,a + b.\)

Measuring the Perimeter of an Equilateral Triangle

\(ABC\) is the equilateral triangle with sides \(AB = BC = AC = a\)

So, the perimeter of \(ABC=\) the length of the sides \( = AB + BC + CA = a + a + a = 3\,a\)

\( = 3 \times {\rm{side}}\)

Units of Perimeter

In CGS system, the unit used for perimeter is \({\rm{cm}}\) and in SI system the unit used is \({\rm{m}}{\rm{.}}\)

Area of a Triangle

The area of a triangle is the region of space enclosed by the three sides of the triangle.

Formula of Area of Triangle

There are several formulas available to calculate the area of a triangle. These are discussed below.

Area of a Triangle when Base and Height are Given

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

The above formula is used when the length of any side and the corresponding height is known or given.

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

Area of a Right Triangle

The above formula is used in right-angled triangles directly.

Hence, the formula used to calculate the area of a triangle is \(\frac{1}{2} \times {\rm{base}} \times {\rm{height = }}\frac{1}{2} \times b \times h,\) where  is the base \(b\) and \(h\) is the height.

Type of Triangles Where This Formula Is Used

Apart from the right-angled triangles, this formula can be used to calculate the area of a triangle in any type of triangle if the length of the base and height are given or can be derived from the given information about the triangle.

Area of Triangle With 3 Sides – Heron’s Formula

This formula is used to calculate the area of a triangle if the length of all three sides is known or given.

According to this formula, the area of a triangle is given by,

area \(=\sqrt{s(s-a)(s-b)(s-c)}\)

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

Area of an Equilateral Triangle

Heron’s formula can be used to derive a special formula applicable to calculate the area of an equilateral triangle.

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

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

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

\( = \sqrt {\frac{{3\,a}}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}} = \sqrt {\frac{{3\,{a^4}}}{{16}}} = \frac{{\sqrt 3 }}{4}{a^2},\) where  is the length of the side of the equilateral triangle.

Hence, the area of an equilateral triangle \(=\frac{{\sqrt3}}{4}\times{a^2}=\frac{{\sqrt3}}{4}\times{({\rm{side}})^2}\)

Area of an Isosceles Triangle

Let the two equal sides of an isosceles triangle \(ABC\) be \(AB = AC = a\) 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 \(ABD,\) it can be written that \(A{D^2} + B{D^2} = A{B^2}.\)

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

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

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

Hence, the area of the isosceles triangle \( = \frac{1}{2} \times {\rm{base}} \times {\rm{height}} = \frac{1}{2} \times BC \times AD\)

\( = \frac{1}{2} \times b \times \sqrt {{a^2} – \frac{{{b^2}}}{4}} \)

Area of a Triangle When Coordinates of Its Vertices Are Given

The area of a triangle can be calculated if the coordinates of the three vertices of a triangle on a Cartesian plane are given.

In the triangle \(ABC,\) shown above, \(A\left( {{x_1},{y_1}} \right),B\left( {{x_2},{y_2}} \right)\) and \(C\left( {{x_3},{y_3}} \right)\) are the coordinates of the vertices of the triangle.

The area of this triangle can be calculated by using the formula,

area \( = \frac{1}{2}\left| {{x_1}\left( {{y_2} – {y_3}} \right) + {x_2}\left( {{y_3} – {y_1}} \right) + {x_3}\left( {{y_1} – {y_2}} \right)} \right|\)

Relation Between Areas of Two Triangles Lying on Same Base & Same Parallels

Here, the line \(BC\) is parallel to the line \(DQ.\) Hence, clearly, \(\Delta ABC\) and \(\Delta PBC\) have the same base \(BC\) and they are lying between the same parallel lines \(BC\) and \(DQ.\)

So, the areas of \(\Delta ABC\) and \(\Delta PBC\) are equal. Therefore, area \({\rm{area}}\,\Delta ABC = {\rm{area}}\,\Delta PBC.\)

Relation Between Area of a Triangle and a Parallelogram Lying on the Same Base and Same Parallels

Interesting relation also exists between the area of a triangle and the area of a parallelogram, if they lie on the same base and they are between the same two parallel lines. It can be proved that the area of the triangle is half the area of the parallelogram if they have the same base and lying between the same two parallel lines.

Here, line \(AB\) is parallel to line \(QD.\) Hence, clearly, \(\Delta ABC\) and the parallelogram \(ABDC\) and \(ABPQ\) have the same base \(AB\) and they are lying between the same parallel lines \(AB\) and \(QD.\)

So, the areas of \(\Delta ABC\) is half the area of parallelogram \(ABDC\) and the parallelogram \(ABPQ,\) as well. Therefore, area\(\Delta ABC = \frac{1}{2} \times \) area of parallelogram \(ABDC\) and \({\rm{area}}\,\Delta ABC = \frac{1}{2} \times {\rm{area}}\,{\rm{of}}\,{\rm{parallelogram}}\,ABPQ.\)

Area of a Triangle Units

In CGS system, unit of area of a triangle is \({\rm{c}}{{\rm{m}}^2}\) and in SI system the unit of area of a triangle is \({{\rm{m}}^2}.\)

Solved Examples

Q.1. Can we construct a triangle with the length of sides \(5\,{\rm{cm}},\,7\,{\rm{cm}}\) and \(14\,{\rm{cm}}?\) Explain the answer.
Ans: No. The sum of the two smaller sides \( = 5 + 7 = 12\,{\rm{cm}},\) which is less than the third side \(14\,{\rm{cm}}{\rm{.}}\) To construct a triangle, the sum of the smaller two side must be greater than the third side.

Q.2. Can we construct a triangle with the internal angles \({\rm{5}}{{\rm{0}}^{\rm{o}}}{\rm{,}}\,{\rm{8}}{{\rm{5}}^{\rm{o}}}\) and \({40^{\rm{o}}}?\) Explain the answer.
Ans: No. The sum of the three internal angles \( = {50^{\rm{o}}} + {85^{\rm{o}}} + {40^{\rm{o}}} = {175^{\rm{o}}},\) which is less than \({180^{\rm{o}}}.\) To construct a triangle, the sum of the internal angles must be equal to \({180^{\rm{o}}}.\)

Q.3. Find the area of a right-angled triangle whose lengths of the sides other than the hypotenuse are \(6\,{\rm{cm}}\) and \(9\,{\rm{cm}}\)
Ans: If the lengths of the sides other than the hypotenuse are \(6\,{\rm{cm}}\) and \(9\,{\rm{cm,}}\) then one of length must be the height and the other length will be the base.
So, the area of the triangle \( = \frac{1}{2} \times {\rm{base}} \times {\rm{height}} = \frac{1}{2} \times 6 \times 9 = 27\,{\rm{c}}{{\rm{m}}^2}.\)

Q.4. Find the area of a triangle whose lengths of the sides are \(5\,{\rm{cm}},\,6\,{\rm{cm}}\), and  \(9\,{\rm{cm}}\)

Ans: Here, the lengths of the three sides are given. So, we shall use Heron’s formula to calculate the area of the triangle.
Here, \(a = 5\;\,{\rm{cm}},\,b = 6\;\,{\rm{cm}}\) and \(c = 9\,{\rm{cm}}\)
According to this formula, the area of a triangle is given by,
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, \(s = \frac{{a + b + c}}{2} = \frac{{5 + 6 + 9}}{2} = \frac{{20}}{2} = 10\,{\rm{cm}}.\)
Hence, the area of the given triangle \( = \sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)} = \sqrt {10\left( {10 – 5} \right)\left( {10 – 6} \right)\left( {10 – 9} \right)} \)
\( = \sqrt {10 \times 5 \times 4 \times 1} = \sqrt {200} = 10\sqrt 2 {\mkern 1mu} \,{\rm{c}}{{\rm{m}}^2}\)

Q.5. Find the perimeter of an equilateral triangle of side \(11\,{\rm{cm}}.\)
Ans: Here, \(a = 11\,{\rm{cm}}\)
Hence, the perimeter of the equilateral triangle with side \(11\,{\rm{cm}} = 3\,a = 3 \times 11 = 33\,{\rm{cm}}.\)

Related Topics to Study

Summary

After going through this article, students will learn different shapes, classifications and properties in a qualitative manner and learn perimeter and area in both qualitative and quantitative manner. Knowing all those, the students will have a reasonably good idea about a triangle and how to use different formulas to obtain its parameter and area.

However, the triangle is a very vast topic. It dominates the entire geometry. This topic would have been very large if everything of a triangle would have been covered here. There is enough scope for a student to go into the detail and depth of each concept discussed here.

FAQs on Triangles

Frequently asked questions related to triangles are listed as follows:

Q. What is a triangle and what are its properties?
A. A triangle is a three-sided polygon that has three angles and three vertices. The main property of a triangle is that the sum of all the angles of a triangle is 180 degrees.

Q. What are different types of triangles?
A. There are in total six different types of triangles, these are acute-angled triangle, right-angled triangle, obtuse-angled triangle, equilateral triangle, scalene triangle and isosceles triangle.

Q. What is an isosceles triangle?
A. An isosceles triangle is a triangle with two equal sides and angles.

Q. What is the formula to calculate the area of a triangle?
A. The formula to calculate the area of a triangle is 1/2 X (Base) X (Height).

Q. What is the formula of perimeter of a triangle?
A. Sum of all three sides of a triangle is the formula of perimeter of a triangle.