ISEE Upper Level Math : Squares

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

Example Question #1 : Squares

Side \(\displaystyle a\) shown below in square \(\displaystyle ABCD\) is equal to 17.5 inches. What is the perimeter of \(\displaystyle ABCD\)?

342px-square_-_geometry.svg

Possible Answers:

\(\displaystyle 306.25\ in\)

\(\displaystyle 70\ in\)

\(\displaystyle 17.5\ in\)

\(\displaystyle 35\ in\)

Cannot be determined

Correct answer:

\(\displaystyle 70\ in\)

Explanation:

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.

\(\displaystyle 17.5\times4=70\)

Example Question #4 : Quadrilaterals

If the area of a square is \(\displaystyle 25x^{2}\), what is the perimeter?

Possible Answers:

\(\displaystyle 4x^{2}\)

\(\displaystyle 5x^{2}\)

\(\displaystyle 20x\)

\(\displaystyle 10x\)

Correct answer:

\(\displaystyle 20x\)

Explanation:

If the area of a square is \(\displaystyle 25x^{2}\), then the length of one side will be equal to the square root of \(\displaystyle 25x^{2}\)

\(\displaystyle \sqrt{25x^{2}}=5x\)

The perimeter is equal to 4 times the length of one side. 

This gives us: \(\displaystyle 5x\cdot 4 = 20x\)

Example Question #1 : Squares

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?

Possible Answers:

\(\displaystyle 12in\)

\(\displaystyle 36in^2\)

\(\displaystyle 240in\)

\(\displaystyle 24in\)

Correct answer:

\(\displaystyle 24in\)

Explanation:

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

\(\displaystyle Perimeter_{square}=4*6in=24 in\)

Example Question #1 : Squares

Inscribed circle

In the above diagram, the circle is inscribed inside the square. The circle has circumference 30. What is the perimeter of the square?

Possible Answers:

\(\displaystyle 60 \pi\)

\(\displaystyle 120 \pi\)

\(\displaystyle \frac{120}{\pi}\)

\(\displaystyle \frac{60}{\pi}\)

Correct answer:

\(\displaystyle \frac{120}{\pi}\)

Explanation:

Call the diameter of the circle \(\displaystyle d\). 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 \(\displaystyle \pi\), so

\(\displaystyle d = \frac{C}{ \pi} = \frac{30}{ \pi}\).

The perimeter of the square is four times this sidelength, so 

\(\displaystyle P= 4d = 4 \cdot \frac{30 }{\pi} = \frac{120}{\pi}\).

Example Question #234 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Find the perimeter of a square with a width of 4cm.

Possible Answers:

\(\displaystyle \text{There is not enough information to solve the problem.}\)

\(\displaystyle 24\text{cm}\)

\(\displaystyle 12\text{cm}\)

\(\displaystyle 8\text{cm}\)

\(\displaystyle 16\text{cm}\)

Correct answer:

\(\displaystyle 16\text{cm}\)

Explanation:

To find the perimeter of a square, we will use the following formula:

\(\displaystyle \text{perimeter of square} = a+b+c+d\)

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

\(\displaystyle \text{perimeter of square} = 4\text{cm} +4\text{cm} +4\text{cm} +4\text{cm}\)

\(\displaystyle \text{perimeter of square} = 16\text{cm}\)

Example Question #1 : Squares

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?

Possible Answers:

\(\displaystyle 104 in\)

\(\displaystyle 676in\)

\(\displaystyle 52 in\)

\(\displaystyle 78 in\)

Correct answer:

\(\displaystyle 104 in\)

Explanation:

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.

\(\displaystyle P=4s=(26in)4=104in\)

 

Example Question #231 : Plane Geometry

Find the perimeter of a square with a width of 18in.

Possible Answers:

\(\displaystyle 72\text{in}\)

\(\displaystyle 64\text{in}\)

\(\displaystyle 36\text{in}\)

\(\displaystyle \text{There is not enough information to solve the problem.}\)

\(\displaystyle 96\text{in}\)

Correct answer:

\(\displaystyle 72\text{in}\)

Explanation:

To find the perimeter of a square, we will use the following formula:

\(\displaystyle P = a+b+c+d\)

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

\(\displaystyle P = 18\text{in} + 18\text{in} + 18\text{in} + 18\text{in}\)

\(\displaystyle P = 72\text{in}\)

Example Question #2 : Squares

Find the perimeter of a square with a side length of \(\displaystyle 3x^2\).

Possible Answers:

\(\displaystyle 9x^2\)

\(\displaystyle 3x^6\)

\(\displaystyle 7x^2\)

\(\displaystyle 3x^8\)

\(\displaystyle 12x^2\)

Correct answer:

\(\displaystyle 12x^2\)

Explanation:

A square has 4 equal sides.

Write the formula for the perimeter of a square.

\(\displaystyle P=4s\)

Substitute the side length.

\(\displaystyle P=4(3x^2)\)

Evaluate the terms on the right.

The answer is:  \(\displaystyle 12x^2\)

Example Question #242 : Isee Upper Level (Grades 9 12) Mathematics Achievement

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?

 

Possible Answers:

Not enough information to solve the problem

\(\displaystyle 60 ft\)

\(\displaystyle 15ft\)

\(\displaystyle 120ft\)

Correct answer:

\(\displaystyle 60 ft\)

Explanation:

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:

\(\displaystyle A=s^2\)

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.

\(\displaystyle 225 ft^2=s^2\)

\(\displaystyle s=\sqrt{225ft^2}=15ft\)

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.

\(\displaystyle P=4s=4(15ft)=60ft\)

Example Question #3 : Squares

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?

Possible Answers:

increased by 20%

decreased by 10%

the area remains the same

decreased by 4%

Correct answer:

decreased by 4%

Explanation:

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.

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