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SAT Math : Circles

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Example Questions

Example Question #401 : Geometry

A park wants to build a circular fountain with a walkway around it. The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide. If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway?

Possible Answers:

$1296\pi \ ft^{3}$

$504\pi \ ft^{3}$

$1936\pi \ ft^{3}$

None of the other answers are correct.

$336\pi \ ft^{3}$

Correct answer:

$504\pi \ ft^{3}$

Explanation:

The following diagram will help to explain the solution:

Foutain

We are searching for the surface area of the shaded region. We can multiply this by the depth (1.5 feet) to find the total volume of this area.

The radius of the outer circle is 44 feet. Therefore its area is 44²π = 1936π. The area of the inner circle is 40²π = 1600π. Therefore the area of the shaded area is 1936π – 1600π = 336π. The volume is 1.5 times this, or 504π.

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Example Question #402 : Geometry

How many times greater is the area of a circle with a radius of 4in., compared to a circle with a radius of 2in.?

Possible Answers:

$4$

$2\pi$

$\pi$

$2$

$4\pi$

Correct answer:

$4$

Explanation:

The area of a circle can be solved using the equation $A=\pi r^{2}$

The area of a circle with radius 4 is $\pi 4^{2}=16\pi$ while the area of a circle with radius 2 is $\pi 2^{2}=4\pi$ . $16\pi \div 4\pi =4$

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Example Question #31 : How To Find The Area Of A Circle

What is the area of a circle whose diameter is 8?

Possible Answers:

32π

16π

64π

8π

12π

Correct answer:

16π

Explanation:

Circarea

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Example Question #321 : Plane Geometry

There are two circles, one with a circumference of $80\pi$ and the other has a circumference of $40\pi$ . What is the ratio of the larger circle's area to the smaller circle's area?

Possible Answers:

$2\pi$

$4\pi$

Correct answer:

Explanation:

The circumference of a circle is equal to the diameter of the circle times $\pi$ . The diameter is equal to twice the radius so:

$C=2\pi r$

The radius of the first can be solved as follows:

$80\pi=2\pi r_{1}$

$r{_{1}}=40$

Likewise for the second circle:

$40\pi=2\pi r_{2}$

$r_{2}=20$

The are of a circle is given by the following formula:

$A=\pi r^2$ .

The ratio of the larger area to the smaller area can be found as follows:

$\frac{A_{1}}{A_{2}}=\frac{\pi r_{1}^{2}}{\pi r_{2}^{2}}=\frac{\pi (40)^2}{\pi(20)^2}=\frac{1600\pi}{400\pi}$

Cancelling out $\pi$ and dividing gives the correct answer of

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Example Question #81 : Circles

In the following diagram, the radius is given. What is area of the shaded region?

Circle_box

Possible Answers:

$16x^{2}$

$32 x^{2} -8\pi x^{2}$

$64 x^{2} - 16\pi x^{2}$

$32 x^{2} - 16\pi x^{2}$

$64 x^{2} -8\pi x^{2}$

Correct answer:

$32 x^{2} -8\pi x^{2}$

Explanation:

This question asks you to apply the concept of area in finding both the area of a circle and square. Since the cirlce is inscribed in the square, we know that its diameter (two times the radius) is the same length as one side of the square. Since we are given the radius, , we can find the area of both the circle and square.

Square:

This gives us the area for the entire square.

The bottom half of the square has area $\frac{1}{2} \cdot 64x^{}2 = 32x^{}2$ .

Now that we have this value, we must find the area that the circle occupies. The area of a circle is given by $\pi \cdot r^{}2$ .

So the area of this circle will be $\pi \cdot 16x^{}2$ .

The bottom half of the circle has half that area:

$\rightarrow \pi \cdot 8x^{}2$

Now that we have both our values, we can subtract the bottom half of the circle from the bottom half of the square to give us the shaded region:

$32 x^{}2 - \pi \cdot 8x^{}2$

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Example Question #31 : How To Find The Area Of A Circle

Find the area of a circle given radius of 7.

Possible Answers:

$49\pi$

$14\pi$

Correct answer:

$49\pi$

Explanation:

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

$A=\pi{r^2}=\pi*7^2=49\pi$

To remember this formula, think about area and how it is in two dimensions, not one or three. So, one of our variables will have to be squared. By process of elimination, it must be r because pi doesn't change so it only makes sense to square the variable that changes between all circles.

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Example Question #332 : Sat Mathematics

The circumference of a circle is $90\pi$ . What is its area?

Possible Answers:

$45\pi$

$2025\pi$

None of the given answers.

$2700\pi$

2025

Correct answer:

$2025\pi$

Explanation:

The circumference C of a circle with radius r is expressed by:

$C=2\pi r$

Our circle's circumference is $90\pi$ . Therefore,

$90\pi = 2 \pi r$

r=45

Plug this r-value into the formula for the area of a circle.

$A=\pi r^2 = \pi \cdot 45^2 = 2025\pi$

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Example Question #33 : How To Find The Area Of A Circle

Target

In the above figure, OA = AB = BC = CD .

What percent of the figure is white?

Possible Answers:

$28 \frac{4}{7} \%$

$37 \frac{1}{2} \%$

$25 \%$

$40 \%$

$33 \frac{1}{3} \%$

Correct answer:

$37 \frac{1}{2} \%$

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi r^{2}$ :

$A _{1}= \pi r^{2} = \pi \cdot 1 ^{2} = \pi$ - this is the area of the white circle in the center.

$A _{2}= \pi r^{2} = \pi \cdot 2^{2} =4 \pi$

$A _{3}= \pi r^{2} = \pi \cdot 3^{2} =9 \pi$

$A _{4}= \pi r^{2} = \pi \cdot 4^{2} =16 \pi$ - this is the total area of the figure.

The area of the white ring is the difference between those at of the second-largest and third-largest circles:

$A _{3} - A_{2} =9 \pi - 4 \pi = 5 \pi$

The total area of the white portion of the figure is

$5 \pi + \pi = 6 \pi$ ,

which is

$\frac{6 \pi}{16 \pi } \times 100 \% = 37 \frac{1}{2} \%$

of the figure.

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Example Question #334 : Geometry

A circle with radius units fits inside of a square such as the circle is tangent to the sides of the square. What is the area inside of the square but outside of the circle?

Possible Answers:

$100\pi$ square units

$100-25\pi$ square units

$25-100\pi$ square units

$25\pi$ square units

$25 - 25\pi$ square units

Correct answer:

$100-25\pi$ square units

Explanation:

If the radius of the circle is units, that means that the diameter is units. Because the circle is tangent to the sides of the square, we also know that each side of the square is also units.

The area of the square is given by its length times its width, or:

$10\cdot10=100$

The area of the circle is given by:

$A=\pi r^2 = \pi \cdot 5^2 = 25\pi$

The area that we're looking for lies outside of the circle but inside of the square. To get this area, we subtract the area of the circle from the area of the square. This gives us our answer of $100-25\pi$ square units.

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Example Question #335 : Geometry

22528212843ba152976c04

The area of the square in this figure is . What is the area of the entire figure given that the semi-circle atop the square is exactly half of a full circle.

Possible Answers:

$36+3\pi$

$36\pi$

None of the given answers.

$36+\frac{9\pi}{2}$

$6\pi$

Correct answer:

$36+\frac{9\pi}{2}$

Explanation:

If the area of the square is , then that means that each side is since the area of a square is given by A=s^2 .

In the figure, the side of the square is equal to the diameter of the semicircle. Therefore, the semicircle's radius is .

The area of the semicircle is given by:

$A= \frac{1}{2}(\pi r^2) = \frac{1}{2}\pi (3^2) = \frac{9\pi}{2}$

The area of the entire figure is the sum of the two component areas.

Our total area is $36 + \frac{9\pi}{2}$ .

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SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #401 : Geometry

Example Question #402 : Geometry

Example Question #31 : How To Find The Area Of A Circle

Example Question #321 : Plane Geometry

Example Question #81 : Circles

Example Question #31 : How To Find The Area Of A Circle

Example Question #332 : Sat Mathematics

Example Question #33 : How To Find The Area Of A Circle

Example Question #334 : Geometry

Example Question #335 : Geometry

All SAT Math Resources

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