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SAT Math : Plane Geometry

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Example Questions

Example Question #7 : How To Find The Area Of A Square

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?

Possible Answers:

144

108

Correct answer:

144

Explanation:

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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Example Question #5 : 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 $\frac{1}{12}$ , what is the length of one side of square WXYZ?

Possible Answers:

$\frac{1}{12}$

$\frac{1}{9}$

$\frac{1}{3}$

$\frac{1}{6}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{3}$

Explanation:

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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Example Question #7 : Squares

Four perfect circles with equal dimensions fit perfectly in a square with a length and width of $4\:cm$ . What is the area of the space inside the square that does not include the circles? (Use 3.14 for the value of $\pi$ .)

Varsity_sat_problems

Possible Answers:

$2.00\:cm^2$

$16.00\:cm^2$

$12.86\:cm^2$

$3.44\:cm^2$

$9.13\:cm^2$

Correct answer:

$3.44\:cm^2$

Explanation:

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 $4\:cm$ , 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 $2\:cm$ and a radius of $1\:cm$ , because 2+2=4 and the square's sides are each $4\:cm$ 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 $1\:cm$ 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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Example Question #9 : Squares

Find the area of a square with side length 4.

Possible Answers:

Correct answer:

Explanation:

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

Substitute the side length of four into the following equation.

Thus,

A=s^2=4^2=16

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Example Question #221 : Plane Geometry

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

Possible Answers:

Correct answer:

Explanation:

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

A=s^2=5^2=25

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Example Question #222 : Plane Geometry

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

Possible Answers:

Cannot be determined

273

Correct answer:

Explanation:

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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Example Question #223 : Plane Geometry

One of the sides of a square on the coordinate plane has an endpoint at the point with coordinates (9, -6 ) ; it has the origin as its other endpoint. What is the area of this square?

Possible Answers:

108

None of the other responses gives a correct answer

117

Correct answer:

117

Explanation:

The length of a segment with endpoints (0,0) and (9, -6 ) 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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Example Question #224 : Plane Geometry

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?

Possible Answers:

None of the given answers.

$25\pi-25$

$25-\frac{25}{4}\pi$

$25-25\pi$

Correct answer:

$25-\frac{25}{4}\pi$

Explanation:

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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Example Question #224 : Geometry

Square 1

Give the area of the provided square in square centimeters.

Possible Answers:

None of these

$180,000 \textrm{ cm} ^{2}$

$360,000 \textrm{ cm} ^{2}$

$1,800 \textrm{ cm} ^{2}$

$3,600 \textrm{ cm} ^{2}$

Correct answer:

$180,000 \textrm{ cm} ^{2}$

Explanation:

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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Example Question #225 : Plane Geometry

A square has the following perimeter:

$350 \textup{ cm}$

Express its area in square meters.

Possible Answers:

$87.5\textrm{ m} ^{2}$

None of these

$0.875\textrm{ m} ^{2}$

$76.5625 \textrm{ m}^{2}$

$0.765625 \textrm{ m}^{2}$

Correct answer:

$0.765625 \textrm{ m}^{2}$

Explanation:

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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SAT Math : Plane Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #7 : How To Find The Area Of A Square

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

Example Question #7 : Squares

Example Question #9 : Squares

Example Question #221 : Plane Geometry

Example Question #222 : Plane Geometry

Example Question #223 : Plane Geometry

Example Question #224 : Plane Geometry

Example Question #224 : Geometry

Example Question #225 : Plane Geometry

All SAT Math Resources

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