Question 969785: Each side of the squares equals to 7units.
How do I calculate the area of an equilateral triangle inside the square with sides 7units?
How do I work out the area of a circle inside a square with sides equal to 7 units?
What would the ratio be of area of triangle:area of circle?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! EQUILATERAL TRIANGLE:
The area of a triangle with sides of lengths  ,  , and  ,
opposite angles of measures  ,  , and  ,
can be calculated as  :
In the case of an equilateral triangle,
all sides have the same length, ,
and all angles measure  ,
so   .
If your equilateral triangle inside a square looks like this  ,
then  and   .
  .
However, it may not be the expected answer,
but you may be able to fit a slightly larger triangle if you "tilt" it,
like this  .
CIRCLE:
The larges circle that you would be able to fit inside a square has a diameter as long as the side of the square:
 If the square side length is  , the circle radius is 
The area of a circle of radius  ,
so a circle of radius  .
In particular, for 
the area would be   .
For the triangle and circle described above,
the ratio of their areas would be
  .
(The side of the square did not matter, as long as we use the same square size to fit the triangle and the circle).
LARGER TRIANGLE:
 In the right triangle corners, according to the Pythagorean theorem,
 and  , so
 , and since  and 
 , and  .
Now, we knew that for a triangle,  ,
and that this tilted triangle has  ,
so  and   .
That makes the area of thte tilted triangle
approximately  ,
and the ratio of areas for the tilted triangle and circle would be
 .
That ratio is approximately  .
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