Solution - Finding an area of a triangle given three sides

Area of Triangle Knowing all Sides :

     Sides: 16.000, 23.000, 10.000
     Area  : 67.3011

Area of Triangle given by its 3 Sides

We will show two ways to find the area. One way is very short - The 2000 years old Heron's Formula. The other method may be longer but it's more "educational" as it teaches us important analytic geometry lessons.

But before we even start, we have to verify that the Basic Triangle Inequality is satisfied.

The Basic Triangle Inequality

The Basic Triangle Inequality states that for a triangle with side lengths a,b,c , the following is true:

     a < b + c,    b < a + c,    c < a + c 

What this means, in plain English, is that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let us now check:
      23.000 < 10.000 + 16.000   true 
      10.000 < 23.000 + 16.000   true 
      16.000 < 23.000 + 10.000   true 

Now that the basic inequalities are satisfied, we know that these three side lengths can make a triangle, so we can move on to calculating the area of said triangle

Heron's Formula for the area of a triangle

The 2000 year old Heron's Formula states that the area of a triangle whose sides have lengths  a, b,  and  c  is

      SQRT (s(s-a)(s-b)(s-c))

where  s  is the semiperimeter of the triangle; that is,

      s = ( a + b + c ) / 2 

Let us calculate
Area =  SQRT ( 24.500 • 1.500 • 14.500 • 8.500 ) =
            SQRT ( 4529.438 ) =
             67.3011  

Find area using the Base Height formula

Let  B  (for base) denote the length of the longest side of the triangle
Let  h  (for height) denote the length of a perpendicular line, from the vertex opposite that side, to the side itself.


Note that  h  splits our triangle into two right-angled triangles, both having  h  as height, the base of the left triangle is denoted by  X  and the base of the right triangle is denoted by  10.000 - X 

To find the area of a triangle, multiply the base by the height, and then divide by 2. In algebraic notation, Area  = 0.5 • 10.000 • h   To be able to find the Area, using this formula, we must know the value of  h 

Using the Pythagorean theorem to find the Height

Applying the Pythagorean theorem to the left right-angled triangle we get:
      h² = (16.000)²- X²  
While the right right-angled triangle is "telling" us that:
      h² = (23.000)²- (10.000 - X)²  

Two things which are equal to  h² , are also equal to one another (It is a property of "Equation" that   if z=p and z=q then p=q ):
      (16.000)²- X² = (23.000)²- (10.000 - X)²  

Expand the above and simplify :
      (10.000)² + (16.000)² - (23.000)² = 20.000 • X 
      -173.000 = 20.000 • X 
      X = -8.650 

Plug this for X in:  h² = (16.000)²- X²  
      h² = (16.000)²- (-8.650)²  
      h² = (181.178) 
      h = sqrt (181.178) = 13.4602  

Put the Triangle Area Formula to use

Finally put the formula  Area = Base * Height * 0.5   to use:
     Area  = 10.000 • 13.460 • 0.5 
     Area  = 67.3011  
Note that this result is identical to the one we got using Heron's Formula !

Area of Triangle Knowing all Sides :

     Sides: 16.000, 23.000, 10.000
     Area  : 67.3011