Radius - ISEE Upper Level Mathematics Achievement

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Question

What is the circumference of a circle with a radius equal to ?

Answer

The circumference can be solved using the following equation:

$C=2r\pi=2(7)\pi=14\pi$

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Question

The perimeter of a given rectangle is equal to the circumference of a given circle. The circle has radius inches; the width of the rectangle is $8 \pi$ inches. What is the length of the rectangle?

Answer

The circumference of a circle with radius inches is

$C = 2 \pi r = 2 \pi \cdot 30 = 60 \pi$ inches.

The perimeter of the rectangle is therefore $60 \pi$ inches. To find its length, substitute $W = 8\pi$ and $P = 60\pi$ into the equation and solve for :

$2 L + 2 \cdot 8 \pi = 60 \pi$

$2 L + 16 \pi = 60 \pi$

$2 L + 16 \pi - 16 \pi = 60 \pi - 16 \pi$

$2 L =44 \pi$

$2 L \div 2=44 \pi \div 2$

$L = 22\pi$ inches

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Question

The radius of a circle is . Give the circumference of the circle in terms of .

Answer

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

$Circumference =2\pi r=2\pi t^2$

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Question

The area of a circle is . Give the circumference of the circle in terms of .

Let $\pi=3.14$ .

Answer

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

$Area=\pi r^2\Rightarrow 25t^2=\pi r^2\Rightarrow r^2=\frac{25t^2}{\pi}$

$\Rightarrow r=\frac{5t}{\sqrt{\pi}}=\frac{5t}{\sqrt{3.14}}\approx 2.82t$

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

$Circumference =2\pi r\Rightarrow Circumference =2\times 3.14\times 2.82t=17.71t$

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Question

What is the circumference of a circle with a radius of ?

Answer

The circumference can be solved using the following equation:

$C=2r\pi$

Where represents the radius. Therefore, when we substitute our radius in we get:

$=2(8)\pi =16\pi$

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Question

The area of the shaded region in the above diagram is $72 \pi$ . Give the circumference of the circle.

Answer

The shaded region is an $80 ^{\circ }$ sector; this represents

$\frac{80}{360} = \frac{80 \div 40 }{360\div 40} = \frac{2}{9}$

of the circle. The area of the region is therefore

$A = \frac{2}{9} \cdot \pi r^{2}$ ,

where is the radius. Setting $A = 72 \pi$ and solving for :

$\frac{2}{9} \cdot \pi r^{2} = 72 \pi$

$\frac{9}{2 \pi} \cdot \frac{2}{9} \cdot \pi r^{2} =\frac{9}{2 \pi} \cdot 72 \pi$

$r^{2} =324$

$r = \sqrt{324 } =18$

The circumference of a circle is its radius multiplied by $2 \pi$ , or

$C = 2 \pi r = 2 \pi \cdot 18 = 36 \pi$

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Question

A circle has a radius of 5 miles, what is its circumference?

Answer

A circle has a radius of 5 miles, what is its circumference?

To find circumference, use the following formula:

$Circumference=2\pi r$

Where r is our radius.

Plug in 5 miles to get our answer:

$Circumference=2\pi (5mi)=10 \pi mi$

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Question

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the circumference of the stamping surface?

Answer

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the circumference of the stamping surface?

Find circumference via the following:

$C=2 \pi r$

Where r is our radius, which is 3.5 cm.

$C=2(3.5cm) \pi =7\pi cm$

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Question

You have a circular window in your vacation room. It has a radius of 9 inches. What is the circumference of the window?

Answer

You have a circular window in your vacation room. It has a radius of 9 inches. What is the circumference of the window?

Begin with the formula for circumference of a circle.

$C=2 \pi r$

Next, plug in our "r" and solve for C

$C= 2 \pi (9in)=18 \pi in$

So, we have our answer!

$18 \pi in$

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Question

Find the circumference of a circle with a radius of 4cm.

Answer

To find the circumference of a circle, we will use the following formula:

$C = 2 \cdot \pi \cdot r$

where r is the radius of the circle.

Now, we know the radius of the circle is 4cm.

Knowing this, we can substitute into the formula. We get

$C = 2 \cdot \pi \cdot 4\text{cm}$

$C = 2\pi \cdot 4\text{cm}$

$C = 8\pi \text{ cm}$

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Question

You have a tomato plant growing in a pot. If the base of the pot has a radius of 6 inches, find the circumference of the footprint of the pot.

Answer

You have a tomato plant growing in a pot. If the base of the pot has a radius of 6 inches, find the circumference of the footprint of the pot.

Begin with the formula for circumference:

$Circ=2 \pi r$

Plug in our 6 inches and simplify:

$C=2 \pi (6in)=12 \pi in$

So, we get:

$12 \pi in$

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Question

Let $\pi = 3.14$

Find the circumference of a circle with a radius of 8in.

Answer

To find the circumference of a circle, we will use the following formula:

$C = 2\pi r$

where r is the radius of the circle.

Now, we know the radius of the circle is 8in.

We also know that $\pi = 3.14$ .

Knowing all of this, we can substitute into the formula. We get

$C = 2 \cdot 3.14 \cdot 8\text{in}$

$C = 16\text{in} \cdot 3.14$

$C = 50.24\text{in}$

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Question

You want to ice skate around the outer edge of a circular pond (the water is frozen, because it is January and you live in Wisconsin). If the pond has a radius of 35 meters, how far will you skate in one lap around the pond?

Answer

You want to ice skate around the outer edge of a circular pond (the water is frozen, because it is January and you live in Wisconsin). If the pond has a radius of 35 meters, how far will you skate in one lap around the pond?

Let's begin by realizing what we are being asked for. They want us to find the total distance we will travel in one lap around the pond. This sounds like circumference, because we are going one full distance around the outside of the circle.

The formula for circumference is as follows.

$C=2 \pi r$

Now, we have r, so just plug in and solve.

$C=2 \pi (35m)=2*35 m \pi=70 \pi m$

So, our answer is:

$70 \pi m$

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Question

You are conducting fieldwork, when you find a tree whose radius at chest height is . What is the circumference of the tree at chest height?

Answer

You are conducting fieldwork, when you find a tree whose radius at chest height is . What is the circumference of the tree at chest height?

The formula for circumference of a circle is:

$C= 2 \pi r$

We have r, so simply plug it in and solve:

$C=2 \pi (1.5m)=3 \pi m$

So, our answer is $3 \pi m$

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Question

Find the circumference of a circle with a radius of 10in.

Answer

To find the circumference of a circle, we will use the following formula:

$C = 2\pi r$

where r is the radius of the circle.

Now, we know the radius of the circle is 10in.

So, we get

$C = 2 \cdot \pi \cdot 10\text{in}$

$C = 20\text{in} \cdot \pi$

$C = 20\pi \text{ in}$

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Question

Find the circumference of a circle with a diameter of 18cm.

Answer

To find the circumference of a circle, we will use the following formula:

$C = \pi d$

where d is the diameter of the circle.

Now, we know the diameter of the circle is 18cm. So, we will substitute. We get

$C = \pi \cdot 18\text{cm}$

$C = 18\pi \text{ cm}$

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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 circumference 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 circumference 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.

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

Next, plug our answer into the circumference formula.

$C=2 \pi r = 2 (13m) \pi =26 \pi m$

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Question

What is the area of a circle with a diameter of , rounded to the nearest whole number?

Answer

The formula for the area of a circle is

$\dpi{100} \pi r^{2}$

Find the radius by dividing 9 by 2:

$\dpi{100} \frac{9}{2}=4.5$

So the formula for area would now be:

$\dpi{100} \pi r^{2}=\pi (4.5)^{2}=20.25\pi \approx 63.6= 64$

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Question

What is the area of a circle that has a diameter of inches?

Answer

The formula for finding the area of a circle is $\pi r^{2}$ . In this formula, represents the radius of the circle. Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius. In order to do this, we divide the diameter by .

$\frac{15}{2}=7.5$

Now we use for in our equation.

$\pi (7.5)^{2}=176.7146 : in^{2}$

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Question

What is the area of a circle with a diameter equal to 6?

Answer

First, solve for radius:

$r=\frac{d}{2}=\frac{6}{2}=3$

Then, solve for area:

$A=r^2\pi=3^2\pi=9\pi$

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