Squares - SAT Math

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Question

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

Answer

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

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Question

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?

Answer

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.

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Question

A half circle has an area of $18\pi$ . What is the area of a square with sides that measure the same length as the diameter of the half circle?

Answer

If the area of the half circle is $18\pi$ , then the area of a full circle is twice that, or $36\pi$ .

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

36π = πr2

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.

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Question

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

Answer

Let S be the original side length. SS would represent the original area. Doubling the side length would give you 2S2S, simplifying to 4(SS), giving a new area of 4x the original, or 144.

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Question

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?

Answer

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 12 : 32 = 1 : 9.

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Question

If the diagonal of a square measures $16\sqrt{2} \ cm$ , what is the area of the square?

Answer

This is an isosceles right triangle, so the diagonal must equal $\sqrt{2}$ times the length of a side. Thus, one side of the square measures $16\ cm$ , and the area is equal to $(16 \ cm)^{2} = 256\ cm^{2}$

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Question

A square $A$ has side lengths of $z$ . A second square $B$ has side lengths of $2.25z$ . How many $A's$ can you fit in a single $B$ ?

Answer

The area of $A$ is $n$ , the area of $B$ is $5.0625n$ . Therefore, you can fit 5.06 $A's$ in $B$ .

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Question

The perimeter of a square is $12\ in.$ If the square is enlarged by a factor of three, what is the new area?

Answer

The perimeter of a square is given by $P=4s=12$ so the side length of the original square is $3\ in.$ The side of the new square is enlarged by a factor of 3 to give $s=9\ in.$

So the area of the new square is given by $A = s^{2} = (9)^{2} = 81 in^{2}$ .

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Question

Give the area of a square with the following perimeter:

$96\textup{ in}$

Answer

The perimeter of a square is equal to the sum of the lengths of its four equally long sides, so the length of one side is one fourth of its perimeter. For this square, this is:

$\frac{96\textrm{ in}}{4} = 24\textrm{ in}$

12 inches are equal to one foot, so divide by 12 to convert to feet:

$\frac{24 \textrm{ in}}{12 \textrm{ in/ft}} = 2 \textrm{ ft}$

The area of a square is equal to the square of the length of one side, so

$A = (2 \textrm{ ft} )^{2} = 4 \textrm{ ft} ^{2}$

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Question

In square WXYZ, point Q is the midpoint of side WZ. If the area of quadrilateral WXYQ is $\frac{1}{12}$ , what is the length of one side of square WXYZ?

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 $\frac{1}{2}s$ , and a height of .

Area of WXYQ = $\frac{s+\frac{1}{2} s}{2}\cdot s=\frac{\frac{3}{2} s}{2}\cdot s =\frac{3}{4} s^2=\frac{1}{12}$

$\frac{4}{3} \cdot \frac{3}{4} s^2=\frac{1}{12} \cdot \frac{4}{3}$

$s^2=\frac{1}{9}$

$s=\frac{1}{3}$

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Question

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 $\pi$ .)

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:

$Area_{square} = 4:cm\cdot 4:cm=16:cm^{2}$

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, $A_{circle}=\pi r^2$ . Substituting in for the value of the radius of one of the circles, we get:

$Area_{circle}=\pi\cdot (1:cm)^{2}$

$Area_{circle}=\pi\cdot 1:cm^2$

$Area_{circle}=\pi:cm^2$

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:

$Area_{four:circles}=4\cdot \pi:cm^2$

$Area_{four:circles}=4\pi:cm^2$

$Area_{square}-Area_{four:circles}=16:cm^2-4\pi:cm^2$

$Area_{square}-Area_{four:circles}=16:cm^2-4(3.14):cm^2$

$Area_{square}-Area_{four:circles}=16:cm^2-12.56:cm^2$

$Area_{square}-Area_{four:circles}= 3.44:cm^2$

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Question

Find the area of a square with side length 4.

Answer

To solve, simply use the formula for the area of a square.

Substitute the side length of four into the following equation.

Thus,

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Question

Find the area of a square whose side length is 5.

Answer

To solve, simply use the formula for the area of a square. Thus,

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Question

A rhombus that is also a rectangle has a side length of . What is its area?

Answer

Remember that a rhombus that is also a rectangle is a square. Knowing that, the calculation is easy: $A_{sq} = s^2 = 9^2 = 81$

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Question

One of the sides of a square on the coordinate plane has an endpoint at the point with coordinates ; it has the origin as its other endpoint. What is the area of this square?

Answer

The length of a segment with endpoints and can be found using the distance formula with $x_{1} = y_{1} = 0$ , $x_{2} = 9$ , $y_{2} = -6$ :

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(9-0)^{2}+(-6-0 )^{2}}$

$= \sqrt{9^{2}+(-6 )^{2}}$

$= \sqrt{81 +36}$

$= \sqrt{117}$

This is the length of one side of the square, so the area is the square of this, or 117.

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Question

A circle is inscribed by a square whose sides are five units long such that the sides of the square barely touch the circle. What is the area inside the square that is not occupied by the circle?

Answer

We want to find the area inside the square that is not taken up by the circle. With that in mind, we can say that the area we're looking for can be expressed by $lw - \pi r^2$ . In other words we are subtracting the area of the circle from the area of the square. This will give us the desired area.

Now, substitute our known values in for the variables to find the desired area:

$lw - \pi r^2 = (5)(5) - \pi (\frac{5}{2})^2 = 25 - \frac{25}{4}\pi$

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Question

Give the area of the provided square in square centimeters.

Answer

The diagonal shown has length 6 meters; to convert to centimeters, multiply by 100:

$6 \textrm{ m } \times 100 \textrm{ cm / m}= 600 \textrm{ cm}$

The easiest way to find the area of the square given the length of a diagonal is to note that since a square is a rhombus, its area is equal to half the product of the lengths of its diagonals. Since one diagonal has length 600 centimeters, so does the other, and the area of the square is therefore

$A = \frac{1}{2} \times 600 \textrm{ cm} \times 600 \textrm{ cm} = 180,000 \textrm{ cm} ^{2}$

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Question

A square has the following perimeter:

$350 \textup{ cm}$

Express its area in square meters.

Answer

One meter comprises 100 centimeters, so divide the perimeter of the sides in centimeters by 100 to obtain the length in meters. This is the same as moving the decimal point left two spaces:

$350. \textrm{ cm } = 3.5 \textrm{ m }$

The perimeter of a square is equal to the sum of the lengths of its four equally long sides, so the length of one side is one fourth of - or, in decimal form, 0.25 times - its perimeter. For this square, this is:

$0.25 \times 3.5\textrm{ m} = 0.875\textrm{ m}$

The area of a square is equal to the square of the length of one side, so

$A = ( 0.875\textrm{ m})^{2} = 0.765625 \textrm{ m}^{2}$

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Question

The area of square R is 12 times the area of square T. If the area of square R is 48, what is the length of one side of square T?

Answer

We start by dividing the area of square R (48) by 12, to come up with the area of square T, 4. Then take the square root of the area to get the length of one side, giving us 2.

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Question

When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?

Answer

Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:

x2 + 64 = (x+2)2

FOIL the right side of the equation.

x2 + 64 = x2 + 4x + 4

Subtract x2 from both sides and then continue with the alegbra.

64 = 4x + 4

64 = 4(x + 1)

16 = x + 1

15 = x

Therefore, the length of the original square is 15 inches.

If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.

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