Why does herons formula work




















We will use some Pythagoras theorem , area of a triangle formula, and algebraic identities to derive Heron's formula. Let us take a triangle having lengths of sides, a, b, and c. Let us assume the side length b is divided into two parts p and q as a perpendicular h falls from the vertex B on the side AC at point M. Consider the triangle below:. Let us begin to calculate the value of h. An equilateral triangle has all sides of the same length. Thus, in this case, the lengths of all sides are equal.

Let us assume the length of all sides is "a", semi-perimeter is "s" and the area of the equilateral triangle is "A". A scalene triangle has all lengths of different sides. Let us assume the length of sides is a, b, c, semi-perimeter is "s" and the area of the scalene triangle is "A". An isosceles triangle has two sides of equal length. Let us assume the length of the two sides is a and one side is b, semi-perimeter is "s" and the area of the isosceles triangle is "A".

We can use Heron's formula to determine the formula for the area of the quadrilateral by dividing it into two triangles. Let us say we have a quadrilateral ABCD with the length of its sides measuring a, b, c, and d. Let us say A and B are joined to show the diagonal of the quadrilateral having length e. The application of Heron's formula in finding the area of the quadrilateral is that it can be used to determine the area of any irregular quadrilateral by converting the quadrilateral into triangles.

Example 1: If the length of the sides of a triangle ABC are 4 in, 3 in, and 5 in. Calculate its area. Find the length of the sides of the triangle. Solution: To find: The length of the sides of the triangle. Example 3: Calculate the area of an isosceles triangle using Heron's formula if the lengths of its sides are 4 units, 8 units, and 8 units. Solution: To find: Area of triangle Given the lengths of sides are 4 units, 8 units, and 8 units.

Solution: Note that we have only been given the lengths of the four sides, but not the length of any diagonal. Heron's formula is used to find the area of the triangle when the lengths of all triangles are given. It can be used to determine areas of different types of triangles, equilateral, isosceles, or scalene triangles.

The heron's formula depends only on the semi-perimeter of a triangle and the length of its three sides. We first determine the value of the semi-perimeter using the lengths of three sides of the triangle. Once the value of the semi-perimeter is obtained we can find the area of the shape. He found the area of the triangle using only the lengths of its sides which made it possible to apply to any type of triangle be it, equilateral, isosceles, or scalene. This formula was further extended by him to calculate areas of quadrilaterals and proved the trigonometric laws such as Laws of cosines or Laws of cotangents.

S stands for semi-perimeter in Heron's formula. It is obtained by halving the value of the perimeter. Yes, we can use Heron's formula in an equilateral triangle. We can derive Heron's formula by using the Pythagoras theorem, area of a triangle formula, and algebraic identities. We construct an altitude from the top vertex to the base of the triangle, which divides the triangle into 2 triangles. Thereby applying the Pythagoras theorem, on both triangles and substituting the values obtained we derive Heron's formula.

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Asked 8 years, 7 months ago. Active 7 years, 11 months ago. Viewed 3k times. I think symbols would help. Less than 1, and the other two sides cannot meet. In fact, there needs to be an additional restriction - that the difference between the two also can't be greater than 1 in magnitude.

I think you have a correct understanding of the reason. But if this is an answer that is being graded, you want to make sure that you get full credit for understanding! Show 1 more comment. Active Oldest Votes. Add a comment. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.



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