Start by working backwards from the area of the circle to find its diameter.

The area of a circle is , and  (you can get this by trial and error pretty quickly, since you know it's between 3 and 4, and since 12.25 ends in a 5, you now that a 5 is involved), so the diameter of the circle is 7. This is also the diagonal of the square. Y

ou may remember that the legs of a  right triangle (of which we have two within the square, once we draw the diagonal) are equal to the hypotenuse divided by . But even if you forget this, you should recall the Pythagorean Theorem that states that . In this case,  and  are equal, so the square of each side is equal to half of , or 24.5. And the square of one side of a square is also equal to the area of the square.

Use the perimeter to find the side length of the square.

Now, use the side length to find the area of the square.

First, use the perimeter to find the side lengths of the square.

Use this information to find the area.

Since we know the lengths of one leg and the hypotenuse, we can calculate the length of the other leg using the Pythagorean Theorem. We can use this form:

Setting  and  equal to the lengths of the hypotenuse and the leg - 13 and 5, respectively:

The perimeter is equal to the sum of the lengths of the three sides:

This is also the perimeter of the square, so the length of each side is one fourth of this, or

The area is the square of this, or

First, we need the radius  of the circle, which can be determined from the area of a circle formula by setting :

Simplifying the expression by splitting the radicand and rationalizing the denominator:

The circumference of the circle is  multiplied by the radius, or

This is also the perimeter of the square, so the length of each side is one fourth of this perimeter, or

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

.

A circle with diameter 10 has radius half this, or 5. The area of the circle can be found using the formula

,

setting :

.

The square also has this area. The length of one side of this square is equal to the square root of this, which is

Simplify this by breaking the radicand, as follows:

The area of a right triangle is equal to half the product of the lengths of its legs. Since we know the lengths of one leg and the hypotenuse, we can calculate the length of the other leg using the Pythagorean Theorem. We can use this form:

Setting  and  equal to the lengths of the hypotenuse and the known leg - 13 and 5, respectively:

The area of the above triangle is

The square also has this area. The length of one side of this square is equal to the square root of this, which is .

The area of Mark's lawn is . The amount of grass seed he needs is  pounds.

He has two options.

Option 1: he can buy three fifty-pound bags for 

Option 2: he can buy two fifty-pound bags and four ten-pound bags for 

The first option is the more economical. 

The area of a rectangle is given by multiplying the width times the height. As a formula:

Where:

is the width and is the height. So we can get:

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. In this problem we have:

or

So the area of the parallelogram would be:

The area of a square is given by:

weher is the side length of a square. In this problem we have , so we can write:

Then:

or: