# SSAT Upper Level Math : How to find the area of a square

## Example Questions

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### Example Question #1 : How To Find The Area Of A Square

The volume of a cube is 1,000 cubic centimeters. Using the conversion factor 2.5 centimeters = 1 inch, give its surface area in square inches, rounding to the nearest square inch.

75 square inches

108 square inches

100 square inches

96 square inches

144 square inches

96 square inches

Explanation:

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.

### Example Question #2 : How To Find The Area Of A Square

The volume of a cube is 64 cubic inches. Find the side length of the cube and its surface area.

Explanation:

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 =

### Example Question #3 : How To Find The Area Of A Square

A square has an area of 16 square inches. Give the diagonal of the square.

Explanation:

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:

### Example Question #4 : How To Find The Area Of A Square

John is going to apply a fertilizer to his farm which has a dimension of 200 feet by 200 feet. Every pound of the fertilizer that he is going to use is sufficient for 40 square feet. If the fertilizer costs 2 dollars per pound, how much he should spend to fertilize his farm?

Explanation:

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.

### Example Question #5 : How To Find The Area Of A Square

The diagonal length of a square is . Find the area of the square in terms of .

Explanation:

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:

### Example Question #31 : Area Of Polygons

A square is inscribed inside a circle that has an area of  square inches. What is the area of the square?

Explanation:

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.

### Example Question #32 : Area Of Polygons

The perimter of a square is  . What is the area of the square?

Explanation:

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

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

### Example Question #33 : Area Of Polygons

The perimeter of a square is  . What is the area of the square?

Explanation:

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

Use this information to find the area.

### Example Question #34 : Area Of Polygons

A square has the same perimeter as the above right triangle. Give the area of the square.

Explanation:

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

### Example Question #35 : Area Of Polygons

The perimeter of a square is equal to the circumference of a circle with area 4. What is the area of the square?

Explanation:

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

.

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