### All SAT Math Resources

## Example Questions

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

ABCD and EFGH are squares such that the perimeter of ABCD is 3 times that of EFGH. If the area of EFGH is 25, what is the area of ABCD?

**Possible Answers:**

25

15

5

225

75

**Correct answer:**

225

Assign variables such that

One side of ABCD = a

and One side of EFGH = e

Note that all sides are the same in a square. Since the perimeter is the sum of all sides, according to the question:

4a = 3 x 4e = 12e or a = 3e

From that area of EFGH is 25,

e x e = 25 so e = 5

Substitute a = 3e so a = 15

We aren’t done. Since we were asked for the area of ABCD, this is a x a = 225.

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

A square has an area of 36. If all sides are doubled in value, what is the new area?

**Possible Answers:**

48

132

144

108

72

**Correct answer:**

144

Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.

### Example Question #1 : Squares

If the perimeter of a square is equal to twice its area, what is the length of one of its sides?

**Possible Answers:**

8

4

3

16

2

**Correct answer:**

2

Area of a square in terms of each of its sides:

Area = S x S

Perimeter of a square:

Perimeter = 4S

So if 'the perimeter of a square is equal to twice its area':

2 x Area = Perimeter

2 x [S x S] = [4S]; divide by 2:

S x S = 2S; divide by S:

S = 2

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

Freddie is building a square pen for his pig. He plans to buy *x* feet of fencing to build the pen. This will result in a pen with an area of *p* square feet. Unfortunately, he only has enough money to buy one third of the planned amount of fencing. Which expression represents the area of the pen he can build with this limited amount of fencing?

**Possible Answers:**

*p*/3

9*p*

*p*/9

*p*/6

3*p*

**Correct answer:**

*p*/9

If Freddie uses *x* feet of fencing makes a square, each side must be *x*/4 feet long. The area of this square is (*x*/4)^{2} = *x*^{2}/16 = *p* square feet.

If Freddie uses one third of *x* feet of fencing makes a square, each side must be *x*/12 feet long. The area of this square is (*x*/12)^{2} = *x*^{2}/144 = 1/9(*x*^{2}/16) = 1/9(*p*) = *p*/9 square feet.

Alternate method:

The scale factor between the small perimeter and the larger perimeter = 1 : 3. Since we're comparing area, a two-dimensional measurement, we can square the scale factor and see that the ratio of the areas is 1^{2 }: 3^{2} = 1 : 9.

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

If the diagonal of a square measures , what is the area of the square?

**Possible Answers:**

**Correct answer:**

This is an isosceles right triangle, so the diagonal must equal times the length of a side. Thus, one side of the square measures , and the area is equal to

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

A square has side lengths of . A second square has side lengths of . How many can you fit in a single ?

**Possible Answers:**

**Correct answer:**

The area of is , the area of is . Therefore, you can fit 5.06 in .

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

The perimeter of a square is If the square is enlarged by a factor of three, what is the new area?

**Possible Answers:**

**Correct answer:**

The perimeter of a square is given by so the side length of the original square is The side of the new square is enlarged by a factor of 3 to give

So the area of the new square is given by .

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

A half circle has an area of . What is the area of a square with sides that measure the same length as the diameter of the half circle?

**Possible Answers:**

108

72

81

144

36

**Correct answer:**

144

If the area of the half circle is , then the area of a full circle is twice that, or .

Use the formula for the area of a circle to solve for the radius:

36π = πr^{2}

r = 6

If the radius is 6, then the diameter is 12. We know that the sides of the square are the same length as the diameter, so each side has length 12.

Therefore the area of the square is 12 x 12 = 144.

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

In square WXYZ, point Q is the midpoint of side WZ. If the area of quadrilateral WXYQ is , what is the length of one side of square WXYZ?

**Possible Answers:**

**Correct answer:**

Let equal the length of one side of the square. Drawing a diagram yields WXYQ is a trapezoid with two right angles, parallel bases of and , and a height of .

Area of WXYQ =

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

Four perfect circles with equal dimensions fit perfectly in a square with a length and width of . What is the area of the space inside the square that does not include the circles? (Use for the value of .)

**Possible Answers:**

**Correct answer:**

Solving for the area not occupied by circles involves subtracting the area of all four circles from the total area of the square, so let's start by calculating the area of the square. We are told that it has a length and width of , so we just need to multiply these together:

Now, we need to calculate the area of one of the circles. Since the four circles fit perfectly into the square, each one has a diameter of and a radius of , because and the square's sides are each long.

Knowing this, we can calculate the area of one of the circles using the equation for the area of a circle, . Substituting in for the value of the radius of one of the circles, we get:

Now, we can multiply our result by to calculate the area encompassed by the four circles together and subtract that value from the area of the square to find the answer: