Lesson Proof of the Heron's formula for the area of a triangle

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Proof of the Heron's formula for the area of a triangle


In this lesson you will learn the proof of the Heron's formula for the area of a triangle:

S = sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29 = ,

where  a,  b  and  c  are the measures of a triangle sides,  S  is the triangle area and  s  is the triangle semiperimeter:  s = %28a+%2B+b+%2B+c%29%2F2.

Let  DELTAABC  be a triangle with the side measures  a,  b  and  c  (Figure 1).

The area of the triangle  DELTAABC  equals half product of the measure of its side            
a = |BC|  and the measure of the altitude  h = |AD|  drawn to this side:

S = a%2Ah%2F2.            (1)

By the  Pythagorean theorem  you have

c%5E2 = h%5E2 + d%5E2   and   b%5E2 = h%5E2 + %28a-d%29%5E2


Figure 1. To the proof of the Heron's formula

in accordance with the  Figure 1.  Distract the second equality from the first one.  You will get   c%5E2 - b%5E2 = 2ad - a%5E2.
Thus you can express  d  via the triangle side measures   d = %28a%5E2+%2B+c%5E2+-+b%5E2%29%2F2a.   Next,  you can express  h  via the triangle side measures
h%5E2 = c%5E2 - d%5E2 = c%5E2 - %28%28a%5E2+%2B+c%5E2+-+b%5E2%29%2F2a%29%5E2 = %28%282ac%29%2F%282a%29%29%5E2 - %28%28a%5E2+%2B+c%5E2+-+b%5E2%29%2F2a%29%5E2 = %28%282ac%29%5E2+-+%28a%5E2+%2B+c%5E2+-+b%5E2%29%5E2%29%2F%284a%5E2%29

                       = = %28b%5E2+-+%28a-c%29%5E2%29%2A%28%28a%2Bc%29%5E2+-b%5E2%29%2F%284a%5E2%29 = %28b-a%2Bc%29%2A%28b%2Ba-c%29%2A%28a%2Bc-b%29%2A%28a%2Bc%2Bb%29%2F%284a%5E2%29

So,   h = %281%2F2a%29%2Asqrt%28%28a%2Bb%2Bc%29%2A%28a%2Bb-c%29%2A%28a%2Bc-b%29%2A%28b%2Bc-a%29%29.   Substitute it into the formula  (1).  You will get for the area of the triangle

S = = sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29,   exactly as the Heron's formula states.


Example 1

Find the area of a triangle with the side measures of  4 cm,  13 cm  and  15 cm.

Solution

The semiperimeter of the triangle is s = %284+%2B13+%2B+15%29%2F2 = 32%2F2 = 16.
The area of the triangle,  according to the Heron's formula,  is

S = sqrt%2816%2A%2816-4%29%2A%2816-13%29%2A%2816-15%29%29 = sqrt%2816%2A12%2A3%2A1%29 = sqrt%2816%2A4%2A9%29 = 8%2A3 = 24 cm%5E2.

Answer.  The area of the triangle is of  24 cm%5E2.


Example 2

Find the measure of the altitude of a triangle with the side measures of  4 cm,  13 cm  and  15 cm  drawn to the shortest side.

Solution

We just have found the area of this triangle in the solution for the  Example 1.  It is of 24 cm%5E2.
From the formula for the area of a triangle   S = ah%2F2  you have for the altitude   h = 2S%2Fa.
Substituting here  S = 24 cm%5E2  and  a = 4 cm  you get  h = 2%2A24%2F4 = 48%2F4 = 12 cm.

Answer.  The altitude of the triangle drawn to its shortest side is of  12 cm.


My other lessons on the topic  Area  in this site are
    - WHAT IS area?
    - Formulas for area of a triangle
    - One more proof of the Heron's formula for the area of a triangle
    - Proof of the formula for the area of a triangle via the radius of the inscribed circle
    - Proof of the formula for the radius of the circumscribed circle
    - Area of a parallelogram
    - Area of a trapezoid
    - Area of a quadrilateral
    - Area of a quadrilateral circumscribed about a circle  and
    - Area of a quadrilateral inscribed in a circle
under the topic  Area and surface area  of the section  Geometry,  and
    - Solved problems on area of triangles
    - Solved problems on area of right-angled triangles
    - Solved problems on area of regular triangles
    - Solved problems on the radius of inscribed circles and semicircles
    - Solved problems on the radius of a circumscribed circle
    - A Math circle level problem on area of a triangle
    - Solved problems on area of parallelograms
    - Solved problems on area of rhombis, rectangles and squares
    - Solved problems on area of trapezoids  and
    - Solved problems on area of quadrilaterals
under the topic  Geometry  of the section  Word problems.

For navigation over the lessons on  Area of Triangles  use this file/link  OVERVIEW of lessons on area of triangles.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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