SOLUTION: 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

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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