Application of Heron’s Formula
In mathematics, Heron's formula is a formula that can be used to determine the area of a triangle, when provided its three side lengths. It can be applied to any shape or kind of triangle, as long as we are familiar with its three side lengths. It is also referred to as Hero’s Formula. Remember that we need not know the angle measurement of a triangle to calculate its area.
As per the Heron, we can calculate the area of any given triangle, whether it is an isosceles, equilateral or scalene, by using the formula, given the sides of the triangle. Let's consider a triangle ABC, whose sides are namely a, b and c, respectively. Therefore, the area of a triangle can be provided as follows: Area of triangle heron's formula = s (s-a) (s-b) (s-c)...√
Where,
S represents semi-perimeter
A represents the length of side a
B represents the length of side b
C represents the length of side c
Heron's Formula for Semi Perimeter of Triangle
The semi perimeter of the triangle by heron's formula is just the perimeter divided by 2: perimeter2.
So, how do we write heron's formula for the semi perimeter of the triangle?
s= \[\frac{\text{Perimeter of triangle}}{2}\] = \[\frac{(a+b+c)}{2}\]
Where,
S represents the semi-perimeter of the triangle is calculated
A represents the length of side a
B represents the length of side b
C represents the length of side c
What Does S Stand for in Heron's Formula?
In heron's formula s stands for semi-perimeter of the triangle. This is used to identify the area and perimeter of the triangle. Heron's formula indicates the area of a triangle whose sides consist of lengths a, b, and c is.
Heron’s Formula for Quadrilateral
Let us learn how to determine the area of a quadrilateral using the Herons formula here.
If PQRS is a quadrilateral, where PQ||RS and PR & QS are the diagonals.
PR divides the quadrilateral PQRS into two triangles PSR and PQR.
Now we obtain two triangles here.
Area of quadrilateral PQRS = Area of ∆PSR + Area of ∆PQR
Thus, if we are familiar with the lengths of all sides of a quadrilateral and the length of diagonal PR, then we can use Heron’s formula to identify the total area.
Therefore, we will first calculate the area of ∆PSR and area of ∆PQR using Heron’s formula and finally, will add them to obtain the final value.
Solved Examples
1. Calculate the area of the triangle whose side’s length measures 12 cm, 15 cm and 21 cm. in addition, identify the length of the altitude on the side which measures 15 cm.
Ans: s = \[\frac{(a+b+c)}{2}\] = \[\frac{(12+15+21)}{2}\] = 24
Area of Triangle= s (s−a) (s−b) (s−c) −−−−−−−−−−√
24×12×9×3−−−−−−−−−√
7776−−−−√ = 88.18square cm
Taking 15 cm as the base length we are required to identify the height
Area, A = \[\frac{1}{2}\] x base x height
\[\frac{1}{2}\] x 15 x h = 88.18
Or h = \[\frac{176.36}{15}\] = 11.75 cm (Rounded to the nearest hundredth).
Fun Facts
Heron's formula is named after a Greek Mathematician and Engineer – Hero of Alexandria, in 10 - 70 AD.
The Greek Mathematician Alexandria also carried forward this idea to calculate the area of quadrilateral and also higher-order polygons.
Heron's formula is used to calculate the area of a triangle using just the 3 side lengths.
The perimeter or area of a triangle by heron’s formula does not rely on the formula for the area that uses base and height.
Heron’s formula also has its wide applications in trigonometry such as giving the law of cosines or law of cotangents, etc.
Tips for the Examination:
Try to solve as many problems as you can from the textbook, then from the exercises given in your classroom and later solve questions from the reference books.
In chapters like mensuration, algebra and trigonometry, learn the formula thoroughly and practice the problems thoroughly.
Do not try to solve problems with a calculator as you will not be allowed to carry a calculator to the exam hall.
You can also take help from any online source, Vedantu has free material of textbooks, solved study material questions, previous year questions etc. You can always download them from the links provided and read them offline or access them from the mobile app or our website.
Try solving questions with your friends, you can study together and solve your doubts together. This will help both the students.
Students can read the chapter before the class which would help them understand the class better.
Go through the resources and figure out important chapters, students are recommended to study all the chapters but practice those chapters important and have comparatively more weightage than other chapters.
Make a small note or flashcards of all the formulas and revise in your free time. Students should realize that recollecting the exact formula is very important in solving the answers.
Students should pay proper attention when they note down the problems, mention the negative and positive symbols carefully. Any slight change in the symbol will alter the whole value.
Students should start with solving problems given in the textbooks, then solve the problems given by the class teacher and then increase the question standard to more complex questions. Solving them and cross-checking the answers with the solutions given will help in improving problem-solving skills and giving full-length mock tests will teach them time management.
FAQs on Heron’s formula
1. What is Heron's Formula for the Equilateral Triangle?
As we are already aware that the equilateral triangle consists of all its sides equal. To determine the area of the equilateral triangle let us first calculate the semi perimeter of the equilateral triangle is:
s = \[\frac{(a+a+a)}{2}\]
s=\[\frac{(3a)}{2}\]
Where ‘a’ is the length of the side.
Now, according to heron’s formula, we know;
Area= s (s−a)(s−b)(s−c)−−−−−−−−−−−−−−√
Since, a = b = c
Hence,
A = √[s(s-a)³]
Which is the needed herons’ formula.
2. What is Heron’s Formula for the Isosceles Triangle?
An isosceles triangle consists of two of its sides equal and the angles corresponding to these sides are incongruence. To determine the area of the isosceles triangle, we can derive the heron’s formula as provided below:
Suppose that ‘a’ be the length of the congruent sides and ‘b’ be the length of the base.
Semi-perimeter (s) = \[\frac{(a+a+b)}{2}\]
S = \[\frac{(2a+b)}{2}\]
Applying the heron’s formula of a triangle,
By plugging the values of sides of an isosceles triangle,
Area = √s(s-a) (s-a) (s-b) = √[s (s-a)² (s-b)]
or
(s-a) √s(s-b)
Which is the needed heron’s formula to identify the area for the provided isosceles triangle.
3. How can we determine the Area of a Quadrilateral using Heron’s Formula?
If we are familiar with the lengths of the sides of a quadrilateral and any one of the diagonal lengths, then by taking diagonal as the common side, we can divide the provided quadrilateral into two triangles and identify the area for both with the help of Heron's formula. Ultimately, we are required to add the areas of the two triangles.
4. What is the difference between Heron’s formula and Pythagoras theorem?
A triangle is a 2-dimensional polygon with three angles, sides and vertices. Heron’s formula is used to find the area of a triangle when we know the sides of a triangle. On the other hand, the Pythagoras theorem deals with a right-angled triangle. In a right-angled triangle, one angle of the triangle measures 90 degrees. It states that the hypotenuse square is equal to the sum of squares of both sides. Heron’s formula can be used to find out the area of any triangle type if we know the length of three sides of a triangle. With the help of the length of the diagonal, we can also find out the area of a quadrilateral. Apart from Heron's formula, we have a standard formula to measure the area of a triangle with the help of the base length, the height of the triangle.
5. What are different types of Triangles and how do we find the area of a triangle with the help of Heron’s formula?
A triangle is a polygon with three sides, three vertices and three angles, it is a two-dimensional representation of a pyramid. Based on the lengths of the sides, a triangle is of three types:
An equilateral triangle has the lengths of all three sides equal to each other. So the angles corresponding to them are also equal and every angle is equal to 60 degrees to fulfil the angle sum property. An isosceles triangle has at least two of its sides with the same length and the angles corresponding to the equal sides are also equal. In a scalene triangle, the lengths of no two sides of a triangle are equal and all the angles are also different.
Based on the angles in a triangle, it is of three types namely, right-angled triangle, obtuse-angled triangle and acute-angled triangle. In a right-angled triangle, at least one angle measures 90 degrees and the Pythagoras theorem can be applied on a right-angled triangle. In an obtuse-angled triangle, at least one angle measures more than 90 degrees but less than 180 degrees. In an acute-angled triangle, at least one angle measures less than 90 degrees. But irrespective of the triangle type, if we know the lengths of the sides, we can find the area of a triangle by applying the following Heron’s formula:
s (s-a) (s-b) (s-c)...√
Here, S denotes the semi-perimeter of the triangle, A, B, C are lengths of the sides of a triangle.
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