ISEE Upper Level Quantitative : Squares
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Example Questions
Example Question #11 : Squares
Square 1 is inscribed inside a circle. The circle is inscribed inside Square 2.
Which is the greater quantity?
(a) Twice the area of Square 1
(b) The area of Square 2
(a) is greater.
(a) and (b) are equal.
It is impossible to tell from the information given.
(b) is greater.
(a) and (b) are equal.
Let 
Therefore, Square 1 has area 
Example Question #12 : Squares
Which is the greater quantity?
(A) The area of a square with sidelength one foot
(B) The area of a rectangle with length nine inches and height fourteen inches
(A) and (B) are equal
(A) is greater
(B) is greater
It is impossible to determine which is greater from the information given
(A) is greater
The area of a square is the square of its sidelength, which here is 12 inches:
The area of a rectangle is its length multiplied by its height, which, respectively, are 9 inches and 14 inches:
The square has the greater area.
Example Question #13 : Squares
A square lawn has sidelength twenty yards. Give its area in square feet.
20 yards converts to 
Example Question #5 : How To Find The Area Of A Square
Rectangle A and Square B both have perimeter 2 meters. Rectangle A has width 25 centimeters. The area of Rectangle A is what percent of the area of Square B?
The perimeter of a rectangle can be given by the formula
Rectangle A has perimeter 2 meters, which is equal to 200 centimeters, and width 25 centimeters, so the length is:
The dimensions of Rectangle A are 75 centimeters and 25 centimeters, so its area is
The sidelength of a square is one-fourth its perimeter, which here is
The area of Rectangle A is therefore
that of Square B.
Example Question #14 : Squares
The sidelength of Square A is three-sevenths that of Square B. What is the ratio of the area of Square B to that of Square A?
None of the other answers give the correct ratio.
Since the ratio is the same regardless of the sidelengths, then for simplicity's sake, assume the sidelength of Square B is 7. The area of Square B is therefore the square of this, or 49.
Then the sidelength of Square A is three-sevenths of 7, or 3. Its area is the square of 3, or 9.
The ratio of the area of Square B to that of Square A is therefore 49 to 9.
Example Question #7 : How To Find The Area Of A Square
Five squares have sidelengths one foot, two feet, three feet, four feet, and five feet.
Which is the greater quantity?
(A) The mean of their areas
(B) The median of their areas
(A) is greater
It is impossible to tell which is greater from the information given
(A) and (B) are equal
(B) is greater
(A) is greater
The areas of the squares are:
Therefore, we are comparing the mean and the median of the set 
The mean of this set is the sum divided by 5:
The median is the middle element after arrangement in ascending order, which is 9.
This makes (A), the mean, greater.
Example Question #6 : How To Find The Area Of A Square
Four squares have sidelengths one meter, 120 centimeters, 140 centimeters, and 140 centimeters. Which is the greater quantity?
(A) The mean of their areas
(B) The median of their areas
(A) and (B) are equal
It is impossible to tell which is greater from the information given
(A) is greater
(B) is greater
(B) is greater
The areas of the squares are:
The mean of these four areas is their sum divided by four:
The median is the mean of the two middle values, or
The median, (B), is greater.
Example Question #9 : How To Find The Area Of A Square
The perimeter of a square is 
None of the other responses gives a correct answer.
The length of one side of a square is one fourth its perimeter. Since the perimeter of the square is 
The area of the square is the square of this sidelength, or
Example Question #5 : How To Find The Area Of A Square
The sidelength of a square is 
The area of a square is the square of its sidelength. Therefore, square 
Example Question #15 : Squares
A diagonal of a square has length 
A square being a rhombus, its area can be determined by taking half the product of the lengths of its (congruent) diagonals:
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