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The Area of an equilateral traingle is equal to the side squared times the square root of 3 quantity over 4. Triangle ABC is equilateral by definition so I will save proving that it is.

Each of the sides are equal to aSIN90 + 2aSIN90, so the area would be calculated in T-SQL something like this:

DECLARE @a DECIMAL SELECT (POWER((@a*2*SIN(90) + @a*SIN(90)), 2)*SQRT(3))/4 ;

The reason the sides are equal to aSIN90 + 2aSIN90 is that perpindicular line from any of the sides of the external equilateral triangle to the vertices of the internal equilateral will bisect the 120 degree angle there making two right triangles from the scalene triangle having an unknown quanitity for the base and the remaining legs 2a and a respectively. The two triangles would have bases (opposite sides) equal to hypotenuse (2a and a respectively) times SIN 90 as the SIN of an angle is equal to the opposite over the hypotenuse.

So there you have it. DE, EF, and FD are equal to 2aSIN90 + aSIN90 because they are composed of the opposite sides of right triangles having hypotenuses of 2a and a (a + continuation segment, and continuation segments respectively). As SINtheta is equal to opposite over hypotenuse, opposite is equal to hypotenuse times SINtheta, theta in these cases being 90.

So there you have it again.

thomasmarshall3@acm.org- July 28, 2008

Article Rating 4 out of 5

Hi Itzik

Triangle ABC is equilateral triangle, therefore, each angle is 60 degrees The area of ABC = ( AB x AC ) x sin (60) / 2 = ( a x a ) x (sqrt 3 / 2) /2 = ( a x a ) x (sqrt 3 / 4)

Now look at triangle ADF Angle FAD = 180 - 60 = 120 degrees The area of ADF = (FA x AD) x sin (120) / 2 = ( 2a x a ) x (sqrt 3 / 4)

Similar to triangle CEF and triangle EBD

Total area of triangle DEF ( a x a ) x (sqrt 3 / 4) + 3 ( 2a x a ) x (sqrt 3 / 4) = 7 (a x a) x (sqrt 3 / 4)

SPC- July 30, 2008

Article Rating 4 out of 5