The Diameter, Radius, and Circumference of Circles

The Diameter, Radius, and Circumference of Circles

Hey guys! Welcome to this video on the Radius, Diameter, and Circumference of a Circle.

Circles have been around (round) since as long as the earth has been around. People were able to see natural circles by observing the moon, the sun, and other various naturally circular shapes.

The first technological invention using a circular shape, however, wasn’t until 3500 B.C, and it was the invention of the potter’s wheel. Then, 300 years later, they were used for the wheels of chariots. As people began to see the value, and use for circular shaped objects, they begin to study circles.

Things like radius, diameter, and circumference are terms that helps us to keep track of various measurements of a circle.

So, now, let’s take a look what each of these measurements represent.

First, let’s define midpoint so you’ll understand what I’m talking about as I reference it. So I’m gonna draw a circle. The midpoint is the exact center of the circle. So, somewhere in here.

Now, let’s look at these other terms.

Radius is the length from the midpoint of the circle to the outer edge of the circle. The radius is represented by the lowercase letter “r.”

Diameter is the full length of the circle running from the edge, through the midpoint, all the way to the other side. That is this whole length right here. The diameter of a circle is represented by the letter “d.”

Now, circumference is the distance around the outside edge of this circle. Circumference is represented by the uppercase letter “C.”

Circumference is comparable to the perimeter of a shape, like a parallelogram. If you were to cut the line of a circle, as if it were a string, and lay it out to measure. This length would be equivalent to the circumference. However, since a circle has a continuous curve, we use the word circumference rather than perimeter to distinguish it.

Now that we’ve looked at what the radius, diameter, and circumference are, let’s look at how to calculate each one.

If someone were to just kinda hand you a piece of paper with a circle on it…. Well, actually, that would be pretty weird.

But let’s say we wanted to find the radius, diameter, and circumference of that circle, and all we have is a ruler.

The easiest thing to start with, would be to take the ruler and measure, from the very center of the circle, the length between the outer edge. That would be the diameter.

Let’s say, that when we measured, we got a length of 9 cm for the diameter. Well, we know that if our radius runs from the midpoint to the outer edge, then all we have to do to find the length of our radius would be to divide the length of the diameter by 2.

So, when we take 9 and divide it by 2 we get a radius length of 4.5 cm.

The formula for the radius can be written as \(r=\frac{d}{2}\), and the formula for diameter can be written as \(d=2r\).

Now to find the circumference of a circle, we will need to use a formula.

The formula for the circumference of a circle is \(C=\pi \times d\), or it can be written as \(C=2\times \pi \times r\). Either one works!

Now, you may be asking, “Well where did pi come from, and why do we all the sudden get the circumference if we multiply said pi by our diameter? Who decided that?” If you are not asking that question… You should, and I’m going to answer it anyways.

Pi is a symbol we use in mathematics to represent the number 3.14. And actually that is just pi rounded to the nearest hundredth. Pi actually has no end, and no predictable pattern. It just keeps going.

However, when you see the symbol \(\pi\), generally (and in our case), 3.14 will suffice.

Pi is not a random number that mathematicians made up, and declared “we will multiply the diameter by the number every time, and call it a circumference.” On the contrary, pi was discovered to be the constant ratio between the circumference and the diameter.

That is why and how we got the formula for the circumference of a circle.

Now, let’s take the circle with the diameter of 9 cm, and radius of 4.5 cm, and calculate the circumference.

I’m going to use the formula with the diameter for this one.

So, circumference equals (I’m just gonna rewrite the formula to help us follow our work), \(C=\pi \times d\), equals pi times diameter. So now all we need to do is to plug in our number for diameter. This is equal to, and also we said pi is equal to 3.14, \(C=(3.14)(9cm)=28.26cm\).

And here’s our answer! Now to practice, try drawing a circle on a piece of paper, and measure your diameter with a ruler. Then, find your radius, and circumference.

I hope that this video has been helpful for you. For further help, be sure to subscribe to our channel by clicking below.

See you guys next time!

Practice Questions

Question #1:

 
Determine the circumference of the circle.
circle with a diameter of 8cm

23.16 cm

24.14 cm

25.12 cm

26.11 cm

Answer:

The circumference of a circle can be calculated using either of the following formulas: \(C=𝜋d\) or \(C=2𝜋r\).

We know that the diameter of the circle is 8 cm, and an approximation for pi is 3.14, so we can plug these values into the formula \(C=𝜋d\). The formula becomes \(C=(3.14)(8)\), which simplifies to 25.12. The circumference of the circle is 25.12 cm.

Question #2:

 
Determine the radius of the circle if the circumference is twenty-three inches. Round your answer to the nearest hundredths.
circle diagram displaying radius

3.66 inches

4.65 inches

3.44 inches

4.76 inches

Answer:

The radius of a circle can be calculated if the circumference is known. We know that the circumference of the circle is twenty-three inches, so we can plug this into the formula \(C=2𝜋r\). We also know that an approximation of pi is 3.14, so the only value we do not know is r, the radius. When C and 𝜋 are plugged into the formula it becomes \(23=2(3.14)r\). This can be simplified to \(23=6.28r\), and then both sides of the equation can be divided by 6.28 in order to isolate the variable r. 23 divided by 6.28 equals 3.66 when rounded to the nearest hundredth. The radius of the circle is 3.66 inches.

Question #3:

 
If C represents circumference, r represents radius, and d represents diameter, which formula is incorrect?

\(d=2r\)
\(C=𝜋d\)
\(C=2𝜋r\)
\(r=𝜋dC\)
Answer:

\(r=𝜋dC\) is incorrect because multiplying pi, times diameter, times circumference does not equal the radius. If the diameter is known, then the radius is simply half the value of d.

Question #4:

 
Bicycles from the 1800s look very different from the bikes we see today. In the photograph below, the bicycle’s back wheel has a radius of 9 inches, and the front wheel has a diameter of 60 inches. Using 3.14 as an approximation for pi, what is the difference between the circumference of the front and back wheel?
Old bicycle

161.88 inches

151.88 inches

171.88 inches

131.88 inches

Answer:

Before comparing the front and back wheel, we need to calculate the circumference of each wheel individually. The circumference of a circle can be found using the formula \(C=2𝜋r\) or \(C=𝜋d\). We know the radius of the back wheel is 9 inches, so we can plug this into the formula \(C=2𝜋r\). The formula becomes \(C=2𝜋(9)\) which simplifies to 56.52. The front tire has a diameter of 60 inches so we can plug this into the formula \(C=𝜋d\). The formula becomes \(C=𝜋(60)\) which simplifies 188.4. Now that we know the circumference of each wheel we can simply subtract 56.52 from 188.4. The difference in the wheel circumferences is 131.88 inches.

Question #5:

 
Lauren is planning her trip to London, and she wants to take a ride on the famous ferris wheel called the London Eye. While researching facts about the giant ferris wheel, she learns that the radius of the circle measures approximately 68 meters. What is the approximate circumference of the ferris wheel? Use 3.14 as an approximation for pi.
London Eye

427 meters

488 meters

407 meters

498 meters

Answer:

The formula \(C=2𝜋r\) can be used to calculate the circumference of the ferris wheel. We can plug in 68 for the radius, and 3.14 as an approximation for pi. The formula \(C=2𝜋r\) becomes \(C=2(3.14)(68)\) which simplifies to 427.04, or approximately 427 meters.

448988

 

by Mometrix Test Preparation | Last Updated: March 31, 2021