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
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Heron’s formula is named after a Greek Mathematician and Engineer – Hero of Alexandria, in 10 – 70 AD.
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The Greek Mathematician Alexandria also carried forward this idea to calculate the area of quadrilateral and also higher-order polygons.
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Heron’s formula is used to calculate the area of a triangle using just the 3 side lengths.
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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.
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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:
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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.
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In chapters like mensuration, algebra and trigonometry, learn the formula thoroughly and practice the problems thoroughly.
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Do not try to solve problems with a calculator as you will not be allowed to carry a calculator to the exam hall.
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You can also take help from any online source, 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.
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Try solving questions with your friends, you can study together and solve your doubts together. This will help both the students.
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Students can read the chapter before the class which would help them understand the class better.
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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.
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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.
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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.
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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.