How to find the area of a square - SSAT Upper Level Quantitative

The volume of a cube is where is the length of one edge, so is the cube root of volume:

A cube has six faces and the surface area of a cube is . So we can write:

Surface area =

In order to determine the length of the diagonal of a square we would use the Pythagorean Theorem. First we should find the side length:

Now, "square" the length of one side and multiply by 2, then take the square root of that number to get the length of the diagonal:

We need to use the Pythagorean Theorem in order to solve this problem. We can write:

where is the diagonal length and is the side length. The diagonal length of the square is , so we can write:

The area of the farm is:

square feet. So the amount of the fertilizer he needs can be calculated as:

pounds

Every pound of the fertilizer costs $2, so he needs to spend dollars.

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.

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

Use this information to find the area.

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

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

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 .

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

.

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

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

The surface area of a cube is six times the square of its sidelength, so we find the sidelength. This is the cube root of volume 1,000, so

centimeters.

To rewrite this as inches, divide by 2.5:

inches

The surface area of the cube in square inches is

square inches.

The volume of a cube is where is the length of one edge, so is the cube root of volume:

A cube has six faces and the surface area of a cube is . So we can write:

Surface area =

In order to determine the length of the diagonal of a square we would use the Pythagorean Theorem. First we should find the side length:

Now, "square" the length of one side and multiply by 2, then take the square root of that number to get the length of the diagonal:

We need to use the Pythagorean Theorem in order to solve this problem. We can write:

where is the diagonal length and is the side length. The diagonal length of the square is , so we can write:

The area of the farm is:

square feet. So the amount of the fertilizer he needs can be calculated as:

pounds

Every pound of the fertilizer costs $2, so he needs to spend dollars.

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.

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

Use this information to find the area.