Diameter - GRE Quantitative Reasoning

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Question

The formula to find the radius of the largest circle that can fit in an equilateral triangle is $\text{Radius }= \frac{S}{2\sqrt{3}}$ , where is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of cm?

Answer

The diameter is

$2\cdot \text{Radius }$

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

$\text{Diameter }= \frac{S}{\sqrt{3}}$ .

Substitute in 5 for S and solve. Diameter = $\frac{5}{\sqrt{3}}$ = 2.89 cm

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Question

Quantity A: The diameter of a circle with area of $81\pi$

Quantity B: The diameter of a circle with circumference of $30\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $81\pi$ . Therefore,

$81\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=30\pi$ , . Therefore, Quantity B is larger.

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Question

Quantity A: The diameter of a circle with area of $109\pi$

Quantity B: The diameter of a circle with circumference of $22\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $109\pi$ . Therefore,

$109\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, $r=\sqrt{109}$ . Since , we know:

$d=2\sqrt{109}$

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=22\pi$ , .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that $\sqrt{121}$ is . Therefore, $\sqrt{109}<11$ . This means that $2 * \sqrt{109} < 22$ . Therefore, Quantity B is larger.

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Question

A circle with an area of $30\pi$ is divided into sectors with areas in a ratio of . What is the area of the largest sector?

Answer

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

$\frac{1}{6}:\frac{2}{6}:\frac{3}{6}$

or

$\frac{1}{6}:\frac{1}{3}:\frac{1}{2}$

The largest sector thus has an area equal to

$\frac{1}{2}30\pi=15\pi$

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Question

A rectangle is inscribed inside of a circle such that every corner touches the edge of the circle. If the area of the rectangle is and the perimeter of the rectangle is , what is the area of the circle in inches squared?

Answer

To find the area of the circle, it is important to know either its diameter or radius. For the geometry described in this problem, this is the same as the diagonal of the rectangle.

However, to find the diagonal of the rectangle, the sides must first be known. They can be found, since the perimeter and area are given:

This system of equation can be solved by substitution:

Followed by:

Note that this gives two possible values for , or , though the one selected is irrelevant; the other value will be the value for .

Knowing these two values, the diagonal can be found; it is the hypotenuse of a right triangle formed by these two lengths:

Since the diagonal is also the diameter, the area of the circle is given by:

$\pi \left(\frac{d}{2}\right)^2=\pi \frac{1681^}{4}=420.25\pi$

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Question

The formula to find the radius of the largest circle that can fit in an equilateral triangle is $\text{Radius }= \frac{S}{2\sqrt{3}}$ , where is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of cm?

Answer

The diameter is

$2\cdot \text{Radius }$

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

$\text{Diameter }= \frac{S}{\sqrt{3}}$ .

Substitute in 5 for S and solve. Diameter = $\frac{5}{\sqrt{3}}$ = 2.89 cm

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $81\pi$

Quantity B: The diameter of a circle with circumference of $30\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $81\pi$ . Therefore,

$81\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=30\pi$ , . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $109\pi$

Quantity B: The diameter of a circle with circumference of $22\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $109\pi$ . Therefore,

$109\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, $r=\sqrt{109}$ . Since , we know:

$d=2\sqrt{109}$

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=22\pi$ , .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that $\sqrt{121}$ is . Therefore, $\sqrt{109}<11$ . This means that $2 * \sqrt{109} < 22$ . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

A circle with an area of $30\pi$ is divided into sectors with areas in a ratio of . What is the area of the largest sector?

Answer

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

$\frac{1}{6}:\frac{2}{6}:\frac{3}{6}$

or

$\frac{1}{6}:\frac{1}{3}:\frac{1}{2}$

The largest sector thus has an area equal to

$\frac{1}{2}30\pi=15\pi$

Compare your answer with the correct one above

Question

A rectangle is inscribed inside of a circle such that every corner touches the edge of the circle. If the area of the rectangle is and the perimeter of the rectangle is , what is the area of the circle in inches squared?

Answer

To find the area of the circle, it is important to know either its diameter or radius. For the geometry described in this problem, this is the same as the diagonal of the rectangle.

However, to find the diagonal of the rectangle, the sides must first be known. They can be found, since the perimeter and area are given:

This system of equation can be solved by substitution:

Followed by:

Note that this gives two possible values for , or , though the one selected is irrelevant; the other value will be the value for .

Knowing these two values, the diagonal can be found; it is the hypotenuse of a right triangle formed by these two lengths:

Since the diagonal is also the diameter, the area of the circle is given by:

$\pi \left(\frac{d}{2}\right)^2=\pi \frac{1681^}{4}=420.25\pi$

Compare your answer with the correct one above

Question

The formula to find the radius of the largest circle that can fit in an equilateral triangle is $\text{Radius }= \frac{S}{2\sqrt{3}}$ , where is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of cm?

Answer

The diameter is

$2\cdot \text{Radius }$

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

$\text{Diameter }= \frac{S}{\sqrt{3}}$ .

Substitute in 5 for S and solve. Diameter = $\frac{5}{\sqrt{3}}$ = 2.89 cm

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $81\pi$

Quantity B: The diameter of a circle with circumference of $30\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $81\pi$ . Therefore,

$81\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=30\pi$ , . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $109\pi$

Quantity B: The diameter of a circle with circumference of $22\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $109\pi$ . Therefore,

$109\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, $r=\sqrt{109}$ . Since , we know:

$d=2\sqrt{109}$

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=22\pi$ , .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that $\sqrt{121}$ is . Therefore, $\sqrt{109}<11$ . This means that $2 * \sqrt{109} < 22$ . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

The formula to find the radius of the largest circle that can fit in an equilateral triangle is $\text{Radius }= \frac{S}{2\sqrt{3}}$ , where is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of cm?

Answer

The diameter is

$2\cdot \text{Radius }$

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

$\text{Diameter }= \frac{S}{\sqrt{3}}$ .

Substitute in 5 for S and solve. Diameter = $\frac{5}{\sqrt{3}}$ = 2.89 cm

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $81\pi$

Quantity B: The diameter of a circle with circumference of $30\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $81\pi$ . Therefore,

$81\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=30\pi$ , . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $109\pi$

Quantity B: The diameter of a circle with circumference of $22\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $109\pi$ . Therefore,

$109\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, $r=\sqrt{109}$ . Since , we know:

$d=2\sqrt{109}$

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=22\pi$ , .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that $\sqrt{121}$ is . Therefore, $\sqrt{109}<11$ . This means that $2 * \sqrt{109} < 22$ . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

The formula to find the radius of the largest circle that can fit in an equilateral triangle is $\text{Radius }= \frac{S}{2\sqrt{3}}$ , where is the length of any one side of the triange.

What is the largest diameter of a circle that can fit inside an equilateral triangle with a perimeter of cm?

Answer

The diameter is

$2\cdot \text{Radius }$

To solve for the largest diameter multiply each side by 2.

The resulting formula for diamenter is

$\text{Diameter }= \frac{S}{\sqrt{3}}$ .

Substitute in 5 for S and solve. Diameter = $\frac{5}{\sqrt{3}}$ = 2.89 cm

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $81\pi$

Quantity B: The diameter of a circle with circumference of $30\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $81\pi$ . Therefore,

$81\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, . Since , we know:

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=30\pi$ , . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

Quantity A: The diameter of a circle with area of $109\pi$

Quantity B: The diameter of a circle with circumference of $22\pi$

Which of the following is true?

Answer

Consider each quantity separately.

Quantity A

Recall that the area of a circle is defined as:

$A=\pi r^2$

We know that the area is $109\pi$ . Therefore,

$109\pi=\pi r^2$

Divide both sides by $\pi$ :

Therefore, $r=\sqrt{109}$ . Since , we know:

$d=2\sqrt{109}$

Quantity B

This is very easy. Recall that:

$C=\pi d$

Therefore, if $C=22\pi$ , .

Now, since your calculator will not have a square root button on it, we need to estimate for Quantity A. We know that $\sqrt{121}$ is . Therefore, $\sqrt{109}<11$ . This means that $2 * \sqrt{109} < 22$ . Therefore, Quantity B is larger.

Compare your answer with the correct one above

Question

A circle with an area of $30\pi$ is divided into sectors with areas in a ratio of . What is the area of the largest sector?

Answer

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

$\frac{1}{6}:\frac{2}{6}:\frac{3}{6}$

or

$\frac{1}{6}:\frac{1}{3}:\frac{1}{2}$

The largest sector thus has an area equal to

$\frac{1}{2}30\pi=15\pi$

Compare your answer with the correct one above

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