Squares - GMAT Quantitative

To find the perimeter of a square given its area, take the square root of the area to find its sidelength; then, multiply that sidelength by 4.

is a perfect square trinomial, since

so its square root is , the sidelength.

Multiply this by 4 to get the perimeter:

Cutting the triangle in half yields a right triangle with the diagonal becoming the hypotenuse and the other two legs being the sides of the square. Using the Pythagorean Theorem, we can solve for the other legs of the triangle.

Since both sides of the square are equal to eachother, , therefore:

To find the area of the square:

with leg being one of the sides

The length of one side of the square is feet, or yards. Square this to get the area in square yards:

square yards.

As a square, is also a rhombus. The area of a rhombus is half the product of the lengths of its diagonals, one of which is . Since the diagonals are congruent, this is equal to half the square of :

The areas of the squares are the squares of the sidelengths, so add the squares of the sidelengths. Since 1 foot is equal to 12 inches and 2 feet are equal to 24 inches, the sum of the areas is:

square inches

The square, the pentagon, and the hexagon have a total of 15 sides, all of which are of equal length; the sum of the lengths is one mile, or 5,280 feet, so the length of one side of any of these polygons is

feet.

The square has area equal to the square of this sidelength:

The regular perimeter has sidelength 60 centimeters and therefore perimeter centimeters. The square has as its sidelength centimeters and area square centimeters.

This problem requires us to find the area of a square. Don't let the story behind it distract you, it is simply an area problem. Use the following equation to find our answer:

is the length of one side of the square; in this case we are told that it is , so we can solve accordingly!

Square has area 49, so each of its sides has as its length the square root of 49, or 7. Each side of Square is therefore a hypotenuse of a right triangle with legs 1 and , so each sidelength, including , can be found using the Pythagorean Theorem:

The square of this, which is 37, is the area of Square .

Square has area 25, so each side has length the square root of 25, or 5.

Specifically, , and, as given, .

Since is a right triangle with hypotenuse and legs and , can be found using the Pythagorean Theorem:

The area of is

Since all four triangles, by symmetry, are congruent, all have this area. the area of Square is the area of Square plus the areas of the four triangles, or .

To make this easier, assume that and - the reasoning generalizes. Then Square has sidelength 15 and area . The sidelength of Square , each side being a hypotenuse of a right triangle with legs 2 and 13, is

.

The square of this, 173, is the area of Square .

The ratio is .

To make this easier, assume that and ; the results generalize.

Each side of Square has length 8, so the area of Square is 64.

Each of the four right triangles has legs 7 and 1, so each has area ; Square has area four times this subtracted from the area of Square , or

.

The area of Square is

of that of Square .

Of the five choices, 80% comes closest.

The formula for the area of a circle is

.

If the area is 100, the radius is as follows:

The circle has circumference times its radius, or

This is also the perimeter of the square, so the sidelength of the square is one-fourth this, or

The area of the square is the square of this, or

A circle with radius 8 has as its circumference times this, or

.

This is also the perimeter of the square, so the sidelength is one fourth of this, or

.

The area is the square of this, or

.

The length of the hypotenuse of a right triangle with legs 8 and 12 can be determined using the Pythagorean Theorem:

Since this is also the perimeter of the square, its sidelength is one fourth of this, or

The area of the square is the square of this sidelength, or

The perimeter of a square, and any shape for that matter, is found by adding up all the exterior sides. Since all sides are equal in a square, we can say:

where represents the length of a side

We can solve for the side length using the information provided:

The area of a square is found by squaring the side length:

The length of one side of a square is the perimeter divided by 4:

Square this to get the area:

The diagonal divides the square into two 45-45-90 triangles, which have side lengths in the ratio of . Therefore, the diagonal is the side length .

There are two pretty easy ways to solve this problem, both of which require you to see that this is actually a triangle question.

The first method is the fastest. If you can recognize that the diagonal of a square makes a triangle, you can then recall that triangles have side lengths with ratios of . Therefore:

The second way to solve this problem is to use Pythagorean Theorem:

where and are leg lengths of a triangle and the length of the hypotenuse. In this case, we want to solve for the diagonal, which is the hypotenuse of the triangle it forms, so we are solving for :

So again, we get .

By definition, all four sides of a square are the same length. So if one side has a length of , the length and width of the square are both . The diagonal of the square is just the hypotenuse of a right triangle with the length and width as the other two sides, so we can use the Pythagorean Theorem to calculate the length of the diagonal: