How to find the length of the 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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