Squares - ISEE Upper Level Quantitative Reasoning
Card 1 of 128
What is the diagonal of a square with a side of 4?
What is the diagonal of a square with a side of 4?
Tap to reveal answer
Squares have all congruent sides. To find the diagonal, first recognize that you're dealing with an isoceles triangle when you draw the diagonal in the square. That means that two of the sides are congruent in the triangle. Thus, it's a special 45-45-90 triangle. In such triangles, the sides are x and the hypotenuse is 
. Since we know x is 4, we can plug in 4 to the expression 
. Thus, the answer is 
.
Squares have all congruent sides. To find the diagonal, first recognize that you're dealing with an isoceles triangle when you draw the diagonal in the square. That means that two of the sides are congruent in the triangle. Thus, it's a special 45-45-90 triangle. In such triangles, the sides are x and the hypotenuse is
← Didn't Know|Knew It →
You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of 
, what will the length of the hypotenuse of the triangles be?
You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of
Tap to reveal answer
You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of 
, what will the length of the hypotenuse of the triangles be?
This problem is trying to distract you by thinking of triangles. What we are really asked to find here is the length of the diagonal of a square with sides of 15 inches.
Splitting a square along its diagonal yields two 45/45/90 triangles. If you know the ratios for 45/45/90 triangles, you can find the answer very quickly.
Think:
Meaning that if the two short sides are x units long, the hypotenuse will be x times the square root of two units long.
In our current case, our short sides are 15 inches long, so our hypotenuse will be
You could also solve this with Pythagorean Theorem.
a and b are both 15 in, so we can solve.
So,our answer is
You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of
This problem is trying to distract you by thinking of triangles. What we are really asked to find here is the length of the diagonal of a square with sides of 15 inches.
Splitting a square along its diagonal yields two 45/45/90 triangles. If you know the ratios for 45/45/90 triangles, you can find the answer very quickly.
Think:
Meaning that if the two short sides are x units long, the hypotenuse will be x times the square root of two units long.
In our current case, our short sides are 15 inches long, so our hypotenuse will be
You could also solve this with Pythagorean Theorem.
a and b are both 15 in, so we can solve.
So,our answer is
← Didn't Know|Knew It →
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?
Tap to reveal answer
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?
To find the diagonal of a square, we can recognize one of two things.
-
The diagonal of a square creates a right triangle, and we can use Pythagorean theorem to find our diagonal.
-
The diagonal of a square creates two 45/45/90 triangles, with side length ratios of 
Using 2), we can find that the diagonal of the square must be 
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?
To find the diagonal of a square, we can recognize one of two things.
-
The diagonal of a square creates a right triangle, and we can use Pythagorean theorem to find our diagonal.
-
The diagonal of a square creates two 45/45/90 triangles, with side length ratios of
Using 2), we can find that the diagonal of the square must be
← Didn't Know|Knew It →
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?
Tap to reveal answer
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?
So, we need to find the diagonal of a square. First, we need to find the side length.
Let's begin with our formula for the area of a square:
where s is our side length and A is our area.
With this formula, we can solve for our side length by plugging in our area and square rooting both sides.
Now, to find the diagonal, we can think of an isosceles right triangle, where the two equal sides are 15 ft. This is also a 45/45/90 triangle, which means the side lengths follow the ratio of 
.
This means our answer is 
.
We could also find this by using Pythagorean Theorem.
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?
So, we need to find the diagonal of a square. First, we need to find the side length.
Let's begin with our formula for the area of a square:
where s is our side length and A is our area.
With this formula, we can solve for our side length by plugging in our area and square rooting both sides.
Now, to find the diagonal, we can think of an isosceles right triangle, where the two equal sides are 15 ft. This is also a 45/45/90 triangle, which means the side lengths follow the ratio of
This means our answer is
We could also find this by using Pythagorean Theorem.
← Didn't Know|Knew It →
One of your holiday gifts is wrapped in a cube-shaped box.
If one of the edges has a length of 6 inches, what is the perimeter of one side of the box?
One of your holiday gifts is wrapped in a cube-shaped box.
If one of the edges has a length of 6 inches, what is the perimeter of one side of the box?
Tap to reveal answer
One of your holiday gifts is wrapped in a cube-shaped box.
If one of the edges has a length of 6 inches, what is the perimeter of one side of the box?
To find perimeter of a square, simply multiply the side length by 4
One of your holiday gifts is wrapped in a cube-shaped box.
If one of the edges has a length of 6 inches, what is the perimeter of one side of the box?
To find perimeter of a square, simply multiply the side length by 4
← Didn't Know|Knew It →
Side 
shown below in square 
is equal to 17.5 inches. What is the perimeter of 
?
Side
Tap to reveal answer
The perimeter of a quadrilateral is the sum of the length of all four sides. In a square, each side is of equal length. Thus, the perimeter is the length of a side (given) times 4.
The perimeter of a quadrilateral is the sum of the length of all four sides. In a square, each side is of equal length. Thus, the perimeter is the length of a side (given) times 4.
← Didn't Know|Knew It →
If the area of a square is 
, what is the perimeter?
If the area of a square is
Tap to reveal answer
If the area of a square is 
, then the length of one side will be equal to the square root of 
.
The perimeter is equal to 4 times the length of one side.
This gives us: 
If the area of a square is
The perimeter is equal to 4 times the length of one side.
This gives us:
← Didn't Know|Knew It →
In the above diagram, the circle is inscribed inside the square. The circle has circumference 30. What is the perimeter of the square?
In the above diagram, the circle is inscribed inside the square. The circle has circumference 30. What is the perimeter of the square?
Tap to reveal answer
Call the diameter of the circle 
. The length of each side of the square also is equal to this.
The diameter of the circle is equal to its circumference divided by 
, so
.
The perimeter of the square is four times this sidelength, so
.
Call the diameter of the circle
The diameter of the circle is equal to its circumference divided by
The perimeter of the square is four times this sidelength, so
← Didn't Know|Knew It →
Find the perimeter of a square with a width of 4cm.
Find the perimeter of a square with a width of 4cm.
Tap to reveal answer
To find the perimeter of a square, we will use the following formula:
where a, b, c, and d are the lengths of the sides of the square.
Now, we know the width of the square has a length of 4cm. Because it is a square, all sides are equal. Therefore, all sides are 4cm.
Knowing this, we can substitute into the formula. We get
To find the perimeter of a square, we will use the following formula:
where a, b, c, and d are the lengths of the sides of the square.
Now, we know the width of the square has a length of 4cm. Because it is a square, all sides are equal. Therefore, all sides are 4cm.
Knowing this, we can substitute into the formula. We get
← Didn't Know|Knew It →
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the perimeter of the square?
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the perimeter of the square?
Tap to reveal answer
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the perimeter of the square?
To find the perimeter of a square, simply multiply the side length by 4. We can do this because perimeter is the distance around the outside of a shape, and squares have 4 equal sides.
While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the perimeter of the square?
To find the perimeter of a square, simply multiply the side length by 4. We can do this because perimeter is the distance around the outside of a shape, and squares have 4 equal sides.
← Didn't Know|Knew It →
Find the perimeter of a square with a width of 18in.
Find the perimeter of a square with a width of 18in.
Tap to reveal answer
To find the perimeter of a square, we will use the following formula:
where a, b, c, and d are the lengths of the sides of the square.
Now, we know the width of the square is 18in. Because it is a square, all sides are equal. Therefore, all sides are 18in.
So, we get
To find the perimeter of a square, we will use the following formula:
where a, b, c, and d are the lengths of the sides of the square.
Now, we know the width of the square is 18in. Because it is a square, all sides are equal. Therefore, all sides are 18in.
So, we get
← Didn't Know|Knew It →
Find the perimeter of a square with a side length of 
.
Find the perimeter of a square with a side length of
Tap to reveal answer
A square has 4 equal sides.
Write the formula for the perimeter of a square.
Substitute the side length.
Evaluate the terms on the right.
The answer is: 
A square has 4 equal sides.
Write the formula for the perimeter of a square.
Substitute the side length.
Evaluate the terms on the right.
The answer is:
← Didn't Know|Knew It →
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the perimeter of her room?
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the perimeter of her room?
Tap to reveal answer
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the perimeter of her room?
So, we need to find the perimeter of a square. First, we need to find the side length.
Let's begin with our formula for the area of a square:
where s is our side length and A is our area.
With this formula, we can solve for our side length by plugging in our area and square rooting both sides.
Now, we are close but not quite done. We need to multiply our side length by 4, because a square always has 4 equal sides.
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the perimeter of her room?
So, we need to find the perimeter of a square. First, we need to find the side length.
Let's begin with our formula for the area of a square:
where s is our side length and A is our area.
With this formula, we can solve for our side length by plugging in our area and square rooting both sides.
Now, we are close but not quite done. We need to multiply our side length by 4, because a square always has 4 equal sides.
← Didn't Know|Knew It →
A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?
A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?
Tap to reveal answer
The area decreases by 20% of 20%, which is 4%.
The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.
10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.
The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.
The area decreases by 20% of 20%, which is 4%.
The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.
10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.
The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.
← Didn't Know|Knew It →
Side 
shown in the diagram of square 
below is equal to 21cm. What is the area of 
?
Side
Tap to reveal answer
To find the area of a quadrilateral, multiply length times width. In a square, since all sides are equal, 
is both the length and width.
To find the area of a quadrilateral, multiply length times width. In a square, since all sides are equal,
← Didn't Know|Knew It →
If Amy is carpeting her living room, which meaures 
feet by 
feet, how many square feet of carpet will she need?
If Amy is carpeting her living room, which meaures
Tap to reveal answer
To find the area of the floor, multiply the length of the room by the width (which is the same forumla used to find the area of a square). The equation can be written: 
Substitute 
feet for 
and 
feet for 
:
Amy will need 
of carpet.
To find the area of the floor, multiply the length of the room by the width (which is the same forumla used to find the area of a square). The equation can be written:
Substitute
Amy will need
← Didn't Know|Knew It →
A rectangle and a square have the same perimeter. The rectangle has length 
centimeters and width 
centimeters. Give the area of the square.
A rectangle and a square have the same perimeter. The rectangle has length
Tap to reveal answer
The perimeter of the rectangle is
centimeters.
This is also the perimeter of the square, so divide this by 
to get its sidelength:
centimeters.
The area is the square of this, or 
square centimeters.
The perimeter of the rectangle is
This is also the perimeter of the square, so divide this by
The area is the square of this, or
← Didn't Know|Knew It →
Four squares have sidelengths 4 inches, 8 inches, 12 inches, and 16 inches. What is the average of their areas?
Four squares have sidelengths 4 inches, 8 inches, 12 inches, and 16 inches. What is the average of their areas?
Tap to reveal answer
The areas of the four squares can be calculated by squaring their sidelengths. Add these areas, then divide by 4:
square inches
The areas of the four squares can be calculated by squaring their sidelengths. Add these areas, then divide by 4:
← Didn't Know|Knew It →
Which of the following is equal to the area of a square with sidelength 
yards?
Which of the following is equal to the area of a square with sidelength
Tap to reveal answer
Multiply the sidelength by 36 to convert from yards to inches:
Square this to get the area:
square inches
Multiply the sidelength by 36 to convert from yards to inches:
Square this to get the area:
← Didn't Know|Knew It →
What is the area of a square in which the length of one side is equal to 
?
What is the area of a square in which the length of one side is equal to
Tap to reveal answer
The area of a square is equal to the product of one side multiplied by another side. Therefore, the area will be equal to:
The next step is to convert the fractions being added together to a form in which they have a common denominator. This gives us:
The area of a square is equal to the product of one side multiplied by another side. Therefore, the area will be equal to:
The next step is to convert the fractions being added together to a form in which they have a common denominator. This gives us:
← Didn't Know|Knew It →