Find the area of a triangle with legs that are 9mm, 6mm, and 12mm

Question

Is it 37mm? 26.1mm? 12.3mm or 11.6mm

Answer 1

Another option for finding the area of any triangle when you know all three sides is to use Heron's formula.

 

First make sure that the numbers a,b,c (sides of the triangle) actually form a triangle. The sum of any two sides must be greater than the 3rd side.

 

Find the semi-perimeter ("s") which is (a+b+c)/2.

 

The by Herons' formula A = sqrt[s(s-a)(s-b)(s-c)].

 

See the following to get the answer for this problem:

 

https://www.desmos.com/calculator/rhmgaha5wn

 

 

Answer 2

The area of a triangle is 1/2 the base (b) length times the height (h), where b is one of the three sides and h is the perpendicular line segment drawn from point opposite from the base to the base. If we choose the 12 mm side as b, then h will divide b into 2 shorter segments, x and 12-x. Since the height segment intersects perpendicularly, that divides the original triangle into 2 smaller right triangles, one with sides 9, h and x and the other with sides 6, h and 12-x. By the Pythagoras theorem (a²+b²=c²) this means

 

91=h²+x² and 36=h²+(12-x)²

 

By modifying one of the equations to give h² in terms of x and then substituting into the other equation, you can derive the value of x. Once you know x, you can determine h and then plug that and 6 (1/2 b) into the equation to determine the area.