The length of one side of a square is the square root of the area - or, equivalently, the area raised to the power of . Here, this is

.

Multiply this by 4 to get the perimeter:

A square courtyard will have paths running diagonally across it (the paths will form an x). If the length of one of the paths will be  long, what will the perimeter of the courtyard be?

To find the perimeter of a square, we need to find the side lengths. We can find the side lengths from the diagonal by recalling that the diagonal of a square creates two 45/45/90 triangles.

A 45/45/90 triangle has the following side-length ratios:  

This means that if our hypotenuse is  long, our other two sides will be 14 meters long

So our side-length is 14 meters, find the perimeter via the following

Draw yourself a picture for this problem.  If you create a square out of two triangles, they meet at the diagonal of the square.  Since, we know the triangles are equilateral, then all of the side lengths of the triangle and square are equal.  Therefore, we just take the side length and add that four times.  

Let  be the sidelength of Square B. Then the sidelength of Square A is , and the areas of Squares A and B are  and , respectively. Since the areas add up to 605 square feet, we solve for  in this equation:

 feet

First we must set up the equation described by the first sentence of the problem. It tells us the perimeter of a square plus its area is equal to . We want to write this in terms of length, so we can write the following:

Where  is the length of one side of the square. We then bring over the  to write the equation as a polynomial that we can factor. This gives us:

A negative length makes no sense, so we can cross out the first result and we're left with the answer, .  We could plug this length back into the original formula and see that a square with this side length would have a perimeter of  and an area of , which add up to , verifying the solution.

Since we know the area of the circle, we can tell that: . Where  is the radius of the circle.

The length of a side of the square will be  since the diameter of the circle is the same length as the side length of the square.

Finally we can calculate the area of the square which will be .  so the area will be , which is our final answer.

The square as an area of 3. From that we can figure out the length of a side of the square, which is the same size as the diameter of the circle.

From the diameter of the circle we can find out, the perimeter of the circle given by the formula: , where  is the length of the diamter.

So  and therefore, the final answer is .

Since BE is half of the diagonal BD, it follows that the size of the diagonal must be .

The length of a diagonal of a square will always be , where  is the length of the side of the square.

Since the diagonal is , the side of the square must be 2.

Let  be the length of one side of the larger square. Then the larger square has area ; the smaller square has area . The area of the region between them, 60, is their difference:

Let  be the length of one side of the smaller square. Then the smaller square has area ; the larger square has area . The area of the region between them, 60, is their difference: