Solution - Finding an area of a triangle given three sides
Step by Step Solution
Area of Triangle Knowing all Sides :
Sides: 12.000, 15.000, 21.000Area : 88.1816
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:
12.000 < 15.000 + 21.000 true
15.000 < 12.000 + 21.000 true
21.000 < 12.000 + 15.000 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 ( 24.000 • 12.000 • 9.000 • 3.000 ) =
SQRT ( 7776.000 ) =
88.1816
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 21.000 - 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 • 21.000 • 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 = (12.000)2- X2
While the right right-angled triangle is "telling" us that:
h2 = (15.000)2- (21.000 - 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 ):
(12.000)2- X2 = (15.000)2- (21.000 - X)2
Expand the above and simplify :
(21.000)2 + (12.000)2 - (15.000)2 = 42.000 • X
360.000 = 42.000 • X
X = 8.571
Plug this for X in: h2 = (12.000)2- X2
h2 = (12.000)2- ( 8.571)2
h2 = (70.531)
h = sqrt (70.531) = 8.3983
Put the Triangle Area Formula to use
Finally put the formula Area = Base * Height * 0.5 to use:
Area = 21.000 • 8.398 • 0.5
Area = 88.1816
Note that this result is identical to the one we got using Heron's Formula !
Area of Triangle Knowing all Sides :
Sides: 12.000, 15.000, 21.000Area : 88.1816
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