Heron’s Formula

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Maths Heron’s Formula

Heron's formula (also known as the Hero's formula) is named after the Hero of Alexandria. It is a formula that can be used to determine the area of a triangle when the length of all the three sides of a triangle are known. It can be to find the area of all the types of a triangle, as long as the length of its three sides is known.


For example, the area of a triangle whose side length is x, y, and z is given by

\[A = \sqrt{s(s - x)(s - y)(s - z)}\]

Where s represents the semi- perimeter of the triangle, i.e.

Semi - Perimeter = x + y + z/2


Heron’s formula can also be written as:

\[A = \frac{1}{4} \sqrt{(x + y + z)(-x + y + z)(x - y + z)(x + y - z)}\]

\[A = \frac{1}{4} \sqrt{2(x^{2}y^{2} + x^{2}z^{2} + y^{2}z^{2}) - (x^{4} + y^{4} + z^{4})}\]

\[A = \frac{1}{4} \sqrt{(x^{2} + y^{2} + z^{2})^{2} - 2(x^{4} + y^{4} + z^{4})}\]

\[A = \frac{1}{4} \sqrt{(x^{2}y^{2} + x^{2}z^{2} + y^{2}z^{2}) -(x^{2} + y^{2} + z^{2})^{2}}\]

\[A = \frac{1}{4} \sqrt{4x^{2}y^{2} - (x^{2} + y^{2}z^{2} - z^{2})^{2}}\]


Heron's Formula of Triangle

Heron's formula is used to calculate the area of a triangle when the length of all the three sides of a triangle are known. It can be applied to all the types of triangles as long as the length of the sides of a triangle is known.


For example, the area of a triangle whose side length is x, y, z is given by

\[Area  = \sqrt{s(s - x)(s - y)(s - z)}\]

Semi Perimeter (s) = Perimeter of Triangle / 2 = x + y + z/2

In the above Heron’s formula of the triangle, ‘s’ represents perimeter, and x, y, and z are three sides of the triangle.


Heron’s Formula for the Area of a Triangle

How to determine the area of a triangle using heron's formula whose sides are x, y, and z respectively?


Following are the steps to determine the area of a triangle whose sides are x,y, and z.

  1. The first step is to determine the semi- perimeter of a triangle i.e. s = (x + y + z) /2.

  2. The second step is to determine the area of a given triangle by applying the heron’s formula i.e. \[A  = \sqrt{s(s - x)(s - y)(s - z)}\]

In the above Heron’s formula for the area of a triangle, ‘s’ represents a semi-perimeter, and x, y, and z are three sides of the triangle.


Heron's Formula for Equilateral Triangle

As we know, all  three sides of an equilateral triangle are equal. Accordingly, to find the area of an equilateral triangle with side 'x', we will first calculate the semi perimeter of an equilateral triangle and that will be:

Semi Perimeter of Equilateral Triangle  = (x + x + x)/2  = 3x/2 ,where x is equal to the length of the sides of an equilateral triangle.

Now, we will find the area of an equilateral triangle using the heron's formula as shown below:

\[Area  = \sqrt{s(s - x)(s - y)(s - z)}\]


As, x = y = z

Accordingly, the area of an equilateral triangle is equal to \[Area  = \sqrt{[s(s - x)^{3}]}\].


Heron's Formula For Isosceles Triangle

An isosceles triangle is a triangle whose two sides are equal and angles corresponding to these sides are congruent to each other. To calculate the area of an isosceles triangle using heron's formula, we will derive the formula as shown below:


Let x be the length of the sides of the congruent sides and y be the length of the base.

Semi perimeter  = x + x + y /2

S = (2x + y/2)


Applying heron’s formula for a triangle, we get:

\[Area  = \sqrt{s(s - x)(s - y)(s - z)}\]

= \[\sqrt{s(s - x)^{2}(s - z)}\]

Or,

\[Area  = (s - x)\sqrt{[s(s - y)]}\]


Hence, the above formula is the required formula to determine the area of a given isosceles triangle.


Heron’s Formula for Quadrilateral

Area of a quadrilateral whose length of the sides, and length of the diagonals are given, can be calculated by dividing the quadrilateral into two triangles and finding the area of both the triangles by using heron’s formula. At last, we need to add the area of both the triangles.


For example, Consider the figure given below:


(Image will be uploaded soon)


How do we calculate the area of quadrilateral ACBD? Simply, we will use Heron’s formula to determine the area of ΔABC and the area of ΔABD, and then add both the triangles to find the total area of a triangle.


Area of ΔABC is calculated as:

Semi - Perimeter of ΔABC = 12 + 25 + 17 / 2 = 54/2 = 27


\[Area = \sqrt{27(27 - 12)(27 - 25)(27 - 17)}\]

\[= \sqrt{27 \times 15 \times 2 \times 10}\]

\[= \sqrt{8100}\]

= 90 sq.units


Area of ΔABD is calculated as:

Semi - Perimeter of ΔABD= 12 + 20 + 28 / 2 = 60/2 =30


\[Area = \sqrt{30(30 - 12)(30 - 20)(30 - 28)}\]

\[= \sqrt{30 \times 18 \times 10 \times 2}\]

\[= \sqrt{10800)}\]

\[= \sqrt{3600 \times 3}\]

\[= 60\sqrt{3}\] sq.units


Total Area of ΔACBD = 90 + 60\[\sqrt{3}\]

\[= 30 (3 + 2\sqrt{3})\] sq. units.

Note, if we were given only the length of the four sides of a quadrilateral, and were not given the length of the diagonal i.e.  AB, we would be unable to calculate the area. It is because, if the length of the diagonal is fixed, there can be an infinite number of quadrilaterals with the same length for the four sides. However, the quadrilateral becomes unique, if its diagonal length is fixed. It states that there can be only one quadrilateral with the given sides and given diagonal length.


Solved Examples

1. Calculate the area of a triangle where the length of every side of a triangle is 5 cm long.

Solution:

Step 1:

Semi - Perimeter (s) =  5 + 5 + 5 / 2 = 7.5


Step 2:

\[Area = \sqrt{7.5(7.5 - 5)(7.5 - 5)(7.5 - 5)}\]

\[= \sqrt{7.5 \times 2.5 \times 2.5 \times 2.5}\]

\[= \sqrt{117.1875}\]

= 10.82 cm²


2. The sides of a triangle are given in the ratio of 3: 5: 7 and its perimeter is 300. Calculate its area.

Solution:

Suppose, the sides of the triangle in meters are 3x, 5x, and 7x.

Hence, we know that perimeter of a triangle is 3x + 5x + 7x = 300

Therefore, 15 x = 300

x = 20

Hence, the sides of a triangle are 3 * 20 m, 5 * 20 m, and 7*20 m i.e. 60 m, 100 m, and 140 m.

Now, we will determine the area of the triangle using heron’s formula as shown below:

We have s = 60 + 100 + 40/2 meters = 150 meters.


And, the area of the triangle will be:

\[A = \sqrt{150(150 - 60)(150 - 100)(150 - 140)}\]

\[= \sqrt{(150)(90)(50)(10)}\] m2

\[= 1500 \sqrt{3}\]m2

FAQ (Frequently Asked Questions)

1. Who Introduced Heron's Formula?

Ans: Heron's formula was introduced by the famous Mathematician and Engineer Heron of Alexandria. The proof of Heron's formula is written in his famous book called Metrica in AD 60. It was introduced to determine the area of the triangle in terms of the length of its sides.

2. Can Heron's Formula be Applied in all Types of the Triangle?

Ans: Heron's can be applied in all types of triangles whether it is equilateral triangle, isosceles or right - angled triangle irrespective of the length of its sides.

3. What is the Use of Heron's Formula?

Ans: We can use Heron's formula to calculate the area of the triangle when the length of all its sides are known. It is also used to determine the area of a quadrilateral by dividing the quadrilateral into two triangular parts.

4. What does the Term ‘s’ Represent in Heron's Formula?

Ans: The term 's' in heron's formula stands for the semi perimeter of a triangle whose area has to be calculated. The semi perimeter of a triangle is equal to the sum of all the three sides of a triangle divided by 2.