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High School Math : Circles

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Example Questions

Example Question #46 : Radius

A circle has the equation below. What is the circumference of the circle?

(x – 2)² + (y + 3)² = 9

Possible Answers:

$16\pi$

$3\pi$

$9\pi$

$6\pi$

Correct answer:

$6\pi$

Explanation:

The radius is 3. Yielding a circumference of $6\pi$ .

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Example Question #1 : How To Find Circumference

Ashley has a square room in her apartment that measures 81 square feet. What is the circumference of the largest circular area rug that she can fit in the space?

Possible Answers:

$10\pi$

$8\pi$

$81\pi$

$18\pi$

$9\pi$

Correct answer:

$9\pi$

Explanation:

In order to solve this question, first calculate the length of each side of the room.

A=s^2

81=s^2

s=9

The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since $C=\pi d$ , the circumference is simply

$C=9\pi$

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Example Question #1 : Diameter And Chords

What is the diameter of a circle with a circumference of $16\pi$ ?

Possible Answers:

Correct answer:

Explanation:

To find the diameter we must understand the diameter in terms of circumference. The equation for the circumference of a circle is $C=D\pi$ , where is the circumference and is the diameter. The circumference is equal to the diameter multiplied by $\pi$ .

We can rearrange $C=D\pi$ to solve for .

$D=\frac{C}{\pi}$

All we have to do is plug in the circumference and divide by $\pi$ , and it will yield the diameter.

$D=\frac{16\pi}{\pi}=16$

The $\pi$ s cancel and the diameter is .

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Example Question #111 : Circles

If the area of a circle is four times larger than the circumference of that same circle, what is the diameter of the circle?

Possible Answers:

Correct answer:

Explanation:

Set the area of the circle equal to four times the circumference πr² = 4(2πr).

Cross out both π symbols and one r on each side leaves you with r = 4(2) so r = 8 and therefore d = 16.

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Example Question #2 : Diameter

The perimeter of a circle is 36 π. What is the diameter of the circle?

Possible Answers:

Correct answer:

Explanation:

The perimeter of a circle = 2 πr = πd

Therefore d = 36

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Example Question #31 : Circles

Sat_math_picture

If the area of the circle touching the square in the picture above is $81\pi$ , what is the closest value to the area of the square?

Possible Answers:

211

162

100

144

Correct answer:

162

Explanation:

Obtain the radius of the circle from the area.

$A=\pi r^2=81\pi$

r^2=81

r=9

Split the square up into 4 triangles by connecting opposite corners. These triangles will have a right angle at the center of the square, formed by two radii of the circle, and two 45-degree angles at the square's corners. Because you have a 45-45-90 triangle, you can calculate the sides of the triangles to be , , and $x\sqrt{2}$ . The radii of the circle (from the center to the corners of the square) will be 9. The hypotenuse (side of the square) must be $9\sqrt{2}$ .

The area of the square is then $(9\sqrt{2})^2=162$ .

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Example Question #51 : Circles

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?

Possible Answers:

$12.5\pi$

$25\pi$

$6\pi$

$6.25\pi$

$5\pi$

Correct answer:

$6.25\pi$

Explanation:

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr².

$A=\pi (5/2)^2=6.25\pi$

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Example Question #1 : How To Find The Ratio Of Diameter And Circumference

What is the ratio of the diameter of a circle to the circumference of the same circle?

Possible Answers:

$\pi$

$D\pi$

$\frac{\pi}{C}$

$\frac{1}{\pi}$

Correct answer:

$\frac{1}{\pi}$

Explanation:

To find the ratio we must know the equation for the circumference of a circle. In this equation, is the circumference and is the diameter.

$C=D\pi$

Once we know the equation, we can solve for the ratio of the diameter to circumference by solving the equation for $\frac{D}{C}$ . We do this by dividing both sides by .

$\frac{C}{\pi}=\frac{D\pi}{\pi}$

$\frac{C}{\pi}=D$

Then we divide both sides by the circumference.

$\frac{C}{\pi C}=\frac{1}{\pi}=\frac{D}{C}$

We now know that the ratio of the diameter to circumference is equal to $\frac{1}{\pi}$ .

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Example Question #1 : How To Find The Ratio Of Diameter And Circumference

What is the ratio of the diameter and circumference of a circle?

Possible Answers:

$\pi$

$2\pi$

$\frac{\pi}{2}$

Correct answer:

Explanation:

To find the ratio we must know the equation for the circumference of a circle is

$Circumference=\pi d$

Once we know the equation we can solve for the ratio of the diameter to circumference by solving the equation for $\frac{diameter}{Circumference}$

we divide both sides by the circumference giving us

$\frac{d}{\pi d}=\frac{1}{\pi }$

We now know that the ratio of the diameter to circumference is equal to .

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Example Question #1 : Diameter And Chords

Let represent the area of a circle and represent its circumference. Which of the following equations expresses in terms of ?

Possible Answers:

C^2

Correct answer:

Explanation:

The formula for the area of a circle is $A=\pi r^2$ , and the formula for circumference is $C=2\pi r$ . If we solve for C in terms of r, we get
$r=C/2\pi$ .

We can then substitute this value of r into the formula for the area:

$A=\pi r^2$

$=\pi (C/2\pi )^2$

$=C^2\pi /4\pi ^2$

$=C^2/4\pi$

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High School Math : Circles

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #46 : Radius

Example Question #1 : How To Find Circumference

Example Question #1 : Diameter And Chords

Example Question #111 : Circles

Example Question #2 : Diameter

Example Question #31 : Circles

Example Question #51 : Circles

Example Question #1 : How To Find The Ratio Of Diameter And Circumference

Example Question #1 : How To Find The Ratio Of Diameter And Circumference

Example Question #1 : Diameter And Chords

All High School Math Resources

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