ISEE Upper Level Quantitative Reasoning - How to find circumference

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

$8 \pi \text{ cm}$

$4\pi \text{ cm}$

$12\pi \text{cm}$

$2\pi \text{ cm}$

$16\pi \text{ cm}$

Explanation

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

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

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

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

Sector

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

$36 \pi$

$18 \pi$

$72 \pi$

$8 \pi$

Explanation

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

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

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?

$26 \pi m$

Explanation

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.

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

Let $\pi = 3.14$

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

$50.24\text{in}$

$25.12\text{in}$

$200.96\text{in}$

$56.52\text{in}$

$43.96\text{in}$

Explanation

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

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

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

$C = 50.24\text{in}$

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

$10\pi mi$

$25 \pi mi$

Explanation

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:

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

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?

$70 \pi m$

$1,225 \pi m$

Not enough information to complete.

Explanation

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.

$C=2 \pi r$