How to find the length of the diameter - ISEE Upper Level Quantitative Reasoning

Card 0 of 36

Question

The area of a circle is $25\pi$ . Give the diameter and radius of the circle.

Answer

The area of a circle can be calculated as $Area=\pi r^2$ where is the radius of the circle, and $\pi$ is approximately .

$Area=\pi r^2\Rightarrow r^2=\frac{Area}{\pi}$

$\Rightarrow r=\sqrt{\frac{Area}{\pi}}=\sqrt{\frac{25 \pi}{\pi}}=\sqrt{25}\Rightarrow r=5$

To find the diameter, multiply the radius by :

$d=2r=2\times 5=10$

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Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $Circumference =2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$Circumference =\pi d=14 \pi\Rightarrow d=\frac{14\pi}{\pi}=14$

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Question

If the value of a radius is $x^{2}-^{\sqrt{x+4}}+2$ , what is the value of the diameter if the value of ?

Answer

If the value of a radius is $x^{2}-^{\sqrt{x+4}}+2$ , and the value of , then the radius will be equal to:

$3^{2}-^{\sqrt{3+4}}+2$

$9-^{\sqrt{7}}+2$

$11-^{\sqrt{7}$

Given that the diameter is twice that of the radius, the diameter will be equal to:

$2(11-^{\sqrt{7}})$

This is equal to:

$22-^{2\sqrt{7}$

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Question

A series of circles has the following radius values:

If the diameter is then calculated for this set, what would be the median diameter?

Answer

The median is the middle number in a set when that set is ordered smalles to largest.

When is ordered smallest to largest, we get

Here, the median would be .

Given that a diameter is twice the radius, the diamater would be (twice the value of ).

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Question

If the area of a circle is equal to $36\pi$ , then what is the diameter?

Answer

If the area of a circle is equal to $36\pi$ , then the radius is equal to .

This is because the equation for the area of a circle is $\pi r^{2}$ .

Thus, $\pi r^{2}=36 \pi$ .

$r^{2}=36$

Then the diameter is 12.

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Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $C=2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$C=2\pi r=\pi d$

$d = \frac{C}{\pi }=\frac{14\pi }{\pi }=14$

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Question

You have a circular lens with a circumference of $16.67 \pi in$ , find the diameter of the lens.

Answer

You have a circular lens with a circumference of $16.67 \pi in$ , find the diameter of the lens.

Begin with the circumference of a circle formula.

$C=2 \pi r$

Now, we know that

Because our radius is half of our diameter.

So, we can change our original formula to be:

$C= \pi d$

Now, we can see that all we need to do is divide our circumference by pi to get our diameter.

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

Now plug in our known and solve:

$\frac{16.67 \pi in}{\pi}=d=16.67 in$

So our answer is 16.67inches

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Question

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the diameter of the crater?

Answer

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the diameter of the crater?

To solve this, we need to recall the formula for the area of a circle.

$A=\pi r^2$

Now, we know A, so we just need to plug in and solve for r!

$169 \pi m^2=\pi r ^2$

Begin by dividing out the pi

Then, square root both sides.

$r=\sqrt{169m^2}=13m$

Now, recall that diameter is just twice the radius.

Compare your answer with the correct one above

Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $C=2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$C=2\pi r=\pi d$

$d = \frac{C}{\pi }=\frac{14\pi }{\pi }=14$

Compare your answer with the correct one above

Question

The area of a circle is $25\pi$ . Give the diameter and radius of the circle.

Answer

The area of a circle can be calculated as $Area=\pi r^2$ where is the radius of the circle, and $\pi$ is approximately .

$Area=\pi r^2\Rightarrow r^2=\frac{Area}{\pi}$

$\Rightarrow r=\sqrt{\frac{Area}{\pi}}=\sqrt{\frac{25 \pi}{\pi}}=\sqrt{25}\Rightarrow r=5$

To find the diameter, multiply the radius by :

$d=2r=2\times 5=10$

Compare your answer with the correct one above

Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $Circumference =2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$Circumference =\pi d=14 \pi\Rightarrow d=\frac{14\pi}{\pi}=14$

Compare your answer with the correct one above

Question

If the value of a radius is $x^{2}-^{\sqrt{x+4}}+2$ , what is the value of the diameter if the value of ?

Answer

If the value of a radius is $x^{2}-^{\sqrt{x+4}}+2$ , and the value of , then the radius will be equal to:

$3^{2}-^{\sqrt{3+4}}+2$

$9-^{\sqrt{7}}+2$

$11-^{\sqrt{7}$

Given that the diameter is twice that of the radius, the diameter will be equal to:

$2(11-^{\sqrt{7}})$

This is equal to:

$22-^{2\sqrt{7}$

Compare your answer with the correct one above

Question

A series of circles has the following radius values:

If the diameter is then calculated for this set, what would be the median diameter?

Answer

The median is the middle number in a set when that set is ordered smalles to largest.

When is ordered smallest to largest, we get

Here, the median would be .

Given that a diameter is twice the radius, the diamater would be (twice the value of ).

Compare your answer with the correct one above

Question

If the area of a circle is equal to $36\pi$ , then what is the diameter?

Answer

If the area of a circle is equal to $36\pi$ , then the radius is equal to .

This is because the equation for the area of a circle is $\pi r^{2}$ .

Thus, $\pi r^{2}=36 \pi$ .

$r^{2}=36$

Then the diameter is 12.

Compare your answer with the correct one above

Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $C=2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$C=2\pi r=\pi d$

$d = \frac{C}{\pi }=\frac{14\pi }{\pi }=14$

Compare your answer with the correct one above

Question

You have a circular lens with a circumference of $16.67 \pi in$ , find the diameter of the lens.

Answer

You have a circular lens with a circumference of $16.67 \pi in$ , find the diameter of the lens.

Begin with the circumference of a circle formula.

$C=2 \pi r$

Now, we know that

Because our radius is half of our diameter.

So, we can change our original formula to be:

$C= \pi d$

Now, we can see that all we need to do is divide our circumference by pi to get our diameter.

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

Now plug in our known and solve:

$\frac{16.67 \pi in}{\pi}=d=16.67 in$

So our answer is 16.67inches

Compare your answer with the correct one above

Question

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the diameter of the crater?

Answer

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the diameter of the crater?

To solve this, we need to recall the formula for the area of a circle.

$A=\pi r^2$

Now, we know A, so we just need to plug in and solve for r!

$169 \pi m^2=\pi r ^2$

Begin by dividing out the pi

Then, square root both sides.

$r=\sqrt{169m^2}=13m$

Now, recall that diameter is just twice the radius.

Compare your answer with the correct one above

Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $C=2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$C=2\pi r=\pi d$

$d = \frac{C}{\pi }=\frac{14\pi }{\pi }=14$

Compare your answer with the correct one above

Question

The area of a circle is $25\pi$ . Give the diameter and radius of the circle.

Answer

The area of a circle can be calculated as $Area=\pi r^2$ where is the radius of the circle, and $\pi$ is approximately .

$Area=\pi r^2\Rightarrow r^2=\frac{Area}{\pi}$

$\Rightarrow r=\sqrt{\frac{Area}{\pi}}=\sqrt{\frac{25 \pi}{\pi}}=\sqrt{25}\Rightarrow r=5$

To find the diameter, multiply the radius by :

$d=2r=2\times 5=10$

Compare your answer with the correct one above

Question

The circumference of a circle is $14\pi$ . Give the diameter of the circle.

Answer

The circumference can be calculated as $Circumference =2\pi r=\pi d$ , where is the radius of the circle and is the diameter of the circle.

$Circumference =\pi d=14 \pi\Rightarrow d=\frac{14\pi}{\pi}=14$

Compare your answer with the correct one above

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