Area of a Triangle

Suppose you need to find out the expense of installing carpet over a triangular floor or you need to lay an extra layer of cheese on a triangular piece of pizza. In order to do so, you need to cover the surface of the triangular floor or pizza. The measurement of this triangular surface is called the area of triangle.


We find the area of a triangle for a lot of purposes. It is useful in architecture, business, academics, farming and everyday life. The concept of the area first came from Mesopotamian, who used to calculate the area to deal with fields and properties. In 2500BC, the architects used the concept of the area of triangle to make the Pyramid at Giza. Scientists like Newton also used the concept of the area of triangle to set the foundation of calculus. Information given below is all you need to know about the area of a triangle.


What is the Area of a Triangle?

What is an Area?

Area is the measurement or the quantitative value of the two dimensional space occupied by an object. In other words, it is the size of the surface of any 2D figure like rectangle, square, triangle, circle, etc.

What is a Triangle?

A triangle is a closed 2D figure having three sides, three vertices and three angles. It is the simplest form of Polygon. A triangle can be formed by joining any three dots such that the line segments connect each other end to end. Three line segments connecting the dots are the sides of the triangle, the point of intersection of two lines is known as vertex and the space between them is called angle. It is also important to know that the sum of all the interior angles of a triangle is always 180 degrees.

Area of triangle as square units

The word ‘area’ stands for the space occupied by a flat object or figure. The area of a triangle is the region enclosed by the sides of a triangle. Let us find the area of a triangle by using square unit areas.  A square unit area is a square having sides of one unit which can either be centimetres or metres. Usually, the length of the side of a unit square is 1cm. To calculate the area of any figure we need to find out how many square units can be adjusted inside the figure without leaving space. In the figure given below we see a triangle, to find out the area of triangle with 3 sides we need to find out how many squares will be needed to cover the triangle.


We can see in the figure given below that 6 full squares of unit length and 4 half triangles can accommodate in the given triangle. These four half squares (triangles) is equal to 2 full squares of the unit area (that is, 1cm2). Thus, in total there are total 8 squares of  1cm2 each. Therefore, the area of the given triangle is 8cm2.

Area of Triangle - Formulas

The total surface or space enclosed by the three boundaries of the triangle is called the area of Triangle. If the base and height of a triangle is given then we can use the area of triangle formula which is given below to find the area.


Area of Triangle = \[\frac{1}{2}\] x length of base x length of height.

                           =  \[\frac{{bh}}{2}\]

The S.I unit is \[{m^2}.\]

The Area Of Different Types Of Triangle.

Classification of Triangles on the basis of sides:

a) Scalene Triangle- It is a triangle having all the sides of different length. The area of scalene triangle of side a, b, c and height h is \[\frac{{hb}}{2}\]


b) Isosceles Triangle - It is a triangle having two sides of equal length. The area of an isosceles triangle of  two equal sides a , base as b and height as h is \[\frac{{hb}}{2}\]


c) Equilateral Triangle - It is a triangle having all the three sides are of equal length.The area of an equilateral triangle with sides a is \[\frac{{\sqrt 3 }}{4}\,{a^2}\].


Classification of Triangles on the basis of angles:

a) Acute Triangle- A triangle in which every angle measures more than zero degrees but less than 90 degrees is called acute-angled Triangle. Area of the acute triangle is more like the area of the scalene triangle. Thus, area of acute triangle of sides  a, b, c and height h is \[\frac{{hb}}{2}\]

                    

b) Obtuse Triangle- A triangle in which one of the angles measures more than 90 degrees but less than 180 degrees is called an obtuse-angled triangle. Area of obtuse triangle of sides  a, b, c and height h is also \[\frac{{hb}}{2}\]

                                             

c) Right Triangle - A triangle in which one of the measures of the angles exactly 90 degrees is called a right triangle. The area of a right triangle of base ‘b’, height ‘h’ and hypotenuse ‘hb’ is  \[\frac{{hb}}{2}\].

                     

Area of a triangle is half the area of the quadrilateral

Below are the few examples of the quadrilateral whose area is double the area of the triangle. 


Proof 1:

When triangle ‘A’ having base 6cm and height 5cm is duplicated and arranged, it actually forms a rectangle of length 6cm and breadth 5cm. Let us check what is the area of a triangle:

We know that the area of rectangle = length x breadth = 30cm square.

So, if the area of triangle is half the area of quadrilateral then the area of the triangle 

=\[\frac{1}{2}\] x length x breadth.

Here, the length and breadth of the quadrilateral is the base and height of the triangle, respectively.


Proof 2:

Triangle B can be doubled, cut into two parts and arranged to make a rectangle.

Area of Triangle = \[\frac{1}{2}\] x base x height

                           = 10 cm square.

Area of Rectangle = length x breadth

                              = 20 cm square.

Thus, the area of the rectangle is double the area of triangle having the same height and base.


Proof 3:

Two triangles ‘C’ can be arranged to make a parallelogram.

Area of Triangle C =\[\frac{1}{2}\] x base x height

                               =  10 cm square

Area of Parallelogram = base x height

                                    = 20 cm square

How to find the area of a triangle?

Given below are the methods to find the area of triangle with pieces of information given. We can find the area of a triangle if :

  1. The length of two sides of a triangle is given.

  2. The length of all the three sides of a triangle is given.

  3. The length of any two sides of a right triangle is given.

  4. The length of one side of the equilateral triangle is given.

  5. The vertices of a triangle on the plane coordinate is given.

  6. The vectors of the triangle are given.


1) When only two sides and an angle of a triangle is given

If two adjacent sides and the angle between the two sides (included angle) is given then the area of the triangle can be found by the formula given below:

Let us suppose the sides of a triangle is named as a, b and c. The angles are named as A, B and C. 

Then, 

Area of triangle  =\[\frac{1}{2}\] ab sinC      

                          = \[\frac{1}{2}\] bc sinA 

                          = \[\frac{1}{2}\]ca sinB



Thus, the area of a triangle is half the product of the length two sides and sin of the angle between the two sides.

Example: Find the area of triangle ABC if side AC = 15 cm, CB = 10 cm and angle C = 25 degree.

              

Solution: 

Area of triangle ABC =\[\frac{1}{2}\] x 15 x 10 x Sin C

                                  = \[\frac{1}{2}\]x 15 x 10 x 0.4226

                                  = 31.695 cm square.

2) When the three sides of a triangle are given

If the length of three sides of a triangle are given then how to calculate the area of a triangle by using Heron’s Formula.

Heron’s Formula:

It states that the area of triangle of sides a, b and c is equal to 

\[A = \sqrt {s(s - a)(s - b)(s - c)} \]

where ‘s’ is the semi-perimeter of the triangle. This formula is used for triangles whose angles are not given and calculation of height is complicated. Thus, Heron’s formula helps us to find the area of a triangle having irregular sides.

Calculation of Semi-perimeter:

The measurement of the semi-perimeter of a triangle having sides a,b and c is important to find the area of the triangle using Heron’s Formula.

Semi-perimeter = \[\frac{{a + b + c}}{2}\]

3) When any two sides of a right-angled triangle are given.

A right angled triangle is a special triangle used as a base of trigonometry, calculus, etc.

One of the three sides of a right-angled triangle itself is height. Thus, there is no need for projection of a perpendicular base from vertex. Moreover, even if the length of height is not given instead,the length of any two sides are given then the length of height can be found by using Pythagorus theorem. As said, right angled-triangle is the base of Trigonometry, so the sides can also be found by using trigonometry formulas if the angles are given.

Area of a right angled triangle = \[\frac{1}{2}\] x base x height.                                           

The height of a right angled triangle can be calculated by using Pythagorus theorems that states: The square of the length hypotenuse (the longest side of right triangle) is equal to the sum of the square of the other two sides (base and perpendicular)




Here, a = base, b = height and c = hypotenuse.

4) When the side of the equilateral triangle is given.

Equilateral triangle is a triangle with all sides equal. To formula to find the area of an equilateral triangle is given below:

A = \[\sqrt{\frac{3}{4}}\] \[s^{2}\]


Here, ‘s’ is the length of the sides of an equilateral triangle.

5) When the vertices of a triangle on the coordinate plane are given

When three vertices of a triangle on the coordinate plane are known then we can do the following check:

  1. If the triangle forms a right-angled triangle then the basic formula of triangle can be used, that is half of the product of height and base.

  2. If it’s not a right triangle then Heron's formula can be used after calculating the semi-perimeter by using the sides of the triangle.

Note: It's not necessary for the triangle to be right angled to use Pythagorus theorem. If we are able to find the height of the triangle in graph then we can calculate the area. Similarly we can use Heron’s formula for any triangle whose three sides are known.


Example: Calculate the area of a triangle ABC whose vertices are (-2,1), (2,4), (4,1).


From the graph we can say that the length of height is 3units and base is 6 units.

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

         =  9 units.


Alternative way: using Matrices

If the three vertices of a triangle are given then we can use the cross product and find the area of a triangle. Suppose if vertices of a triangle are (x1,y1) , (x2,y2) and (x3,y3). Then,

Area of the triangle is, 

6) When the vector sides of triangle are given

If the sides of a triangle is given in vectors then the area will be half the magnitude of their cross products. 

Area of the triangle =  \[\frac{1}{2}|a \times b|\]


Summary

              The area of a triangle

            Formula   

When height(h) and breadth(b) are given

              \[A = \frac{{bh}}{2}\]

When three sides of different lengths (a,b,c) are given

\[A = \sqrt {s(s - a)(s - b)(s - c)} \]

When two sides (a, b) and one angle (C) are given

          \[A = \,\frac{1}{2}ab\,\sin \,C\]

When the side (s) of an equilateral triangle is given

            \[A = \,\frac{{\sqrt 3 }}{4}\,{s^2}\]

When the vertices (x,y,z) of a triangle is given


When two sides of a right angled triangle are given

  \[{H^2} = {P^2} + {B^2}\]


FAQs (Frequently asked questions)

What is the base of a Triangle?

Base of a triangle can be any selected side of a triangle, usually the bottom of a triangle is taken as base. For an isosceles triangle, the unequal side of the triangle is taken as base. Even for the right-angled triangle one of the sides containing right angle (90degrees) can be taken as base.

What is the height of a triangle?

Height of a triangle is the straight line drawn from the top vertex (opp to base) to the base of the triangle, such that the line touches the base perpendicularly making an angle 90 degree.

How to find the length of the height of a triangle?

If the sides of an isosceles or equilateral triangle is given then by Pythagorus theorem we can find the height of the triangle. Incase of a right-angled triangle also, height is one of the sides of a triangle so by using Pythagorus theorem we can find the length of the third side.

Why the area of Triangle is \[\frac{1}{2}\] x base x height?

We know that every quadrilateral can be divided into two or more triangles. Similarly, every triangle is a part of a quadrilateral. Surprisingly, it was found that the area of triangles is equal to half the area of  quadrilateral with the base and height same as that of the triangle.