SOLUTION: If a square and a rhombus stands on the same base ,then the ratio of the areas of the square and the rhombus is????

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Question 632041: If a square and a rhombus stands on the same base ,then the ratio of the areas of the square and the rhombus is????
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the area of a quadrilateral is equal to b*h where b is the base and h is the height.
the area of the square is equal to b*b = b^2 because the base and the height of the square are the same.
the area of the rhombus is equal to b*h.
b is equal to the base.
h is equal to the height.
the height of the rhombus depends on the acute angle that the side of the rhombus makes with the base.
this affects the area of the rhombus.
a larger acute angle (90 degrees is the largest) results in a larger area.
this means that the greaer area is when the rhombus is in fact a square.
anything less than that will result in a smaller area.
assuming that the angle is A, then the height of the rhombus is equal to b*sin(A)
where b is the length of a side of the rhombus.
the area of the rhombus is therefore equal to b*b*sin(A) which is equal to b^2*sin(A).
the ratio of the area of the square to the area of the rhombus is equal to:
area of square / area of rhombus = b^2 / b^2*sin(A) is equal to 1/sin(a) which is equal to cosecant(A).
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a couple of examples:
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example 1:
side of square and rhombus is equal to 5.
angle of rhombus is equal to 90 degrees.
this makes the rhombus a square as well.
area of square is equal to 5*5 = 25
area of rhombus is equal to 5*5*sin(90) which is equal to 5*5*1 which is equal to 25
the areas are the same since the rhombus is also a square.
the ratio is 1:1
sin(90) = 1 which makes cosecant(90) equal to 1/1 which is also equal to 1.
the ratio is equal to the cosecant(90) which is equal to 1 which is the same as a ratio of 1:1.
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example 2:
side of square and rhombus is equal to 5.
angle of rhombus is equal to 45 degrees.
area of square is equal to 5*5 = 25
area of rhombus is equal to 5*5*sin(45) which is equal to 5*5*sqrt(2)/2 which is equal to 25*sqrt(2)/2.
the ratio of the area of the square to the area of the rhombus is equal to 25/(25*sqrt(2)/2) which is equal to 1/(sqrt(2)/2).
this is equal to the cosecant of 45 degrees.
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bottom line is that the ratio of the area of a square to the area of a rhombus with sides that are the same length as the square is equal to the cosecant of the acute angle that the side of the rhombus makes with its base.
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the following diagram show you what i mean:
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