Radius & diameter from circumference | High School Geometry | High School Math | Khan Academy

Summary

This educational video explains how to determine the radius and diameter of a circle given its circumference. It uses the formula C = 2πr to show that if the circumference is 49π, the radius is 24.5 units. Similarly, if the circumference is 1600π, the diameter is 1600 units, highlighting the fundamental role of pi in these calculations.

Takeaways

• 📏 The circumference of a circle is given as 49 pi, and the task is to find the radius.
• 🔍 A circle is drawn to visualize the problem, emphasizing the relationship between the radius and the circumference.
• 📐 The formula for the circumference of a circle is 2πr, where r is the radius.
• 🔢 Pi (π) is defined as the ratio of the circumference to the diameter of a circle.
• 🔍 The diameter of a circle is twice the radius (2r), and thus the circumference is π × 2r.
• 🔧 To solve for the radius, substitute the given circumference (49 pi) into the formula and solve for r.
• 🧩 After substituting, the equation becomes 49π = 2πr, and dividing both sides by 2π gives r = 24.5.
• 🌐 Another example is given with a circumference of 1600 pi, and the task is to find the diameter.
• 🔗 The relationship between the circumference and the diameter is used again, with the formula C = πD, where C is the circumference and D is the diameter.
• 📉 By dividing 1600 pi by pi, the diameter is found to be 1600 units, assuming the units are consistent.
• 🌟 Circles and the concept of pi are highlighted as fundamental and recurring in mathematics.

Q & A

What is the given circumference of the first circle mentioned in the transcript?

The given circumference of the first circle is 49 pi.

How is the circumference of a circle related to its radius?

The circumference of a circle is equal to two pi times the radius (C = 2πr).

What equation is used to find the radius when the circumference is known?

The equation used is C = 2πr. By substituting the known circumference, we can solve for the radius.

How is the radius calculated from the given circumference of 49 pi?

By setting 49 pi equal to 2πr and dividing both sides by 2π, the radius is found to be 24.5 units.

What is the diameter of a circle in terms of its radius?

The diameter of a circle is twice the radius (d = 2r).

What is the given circumference of the second circle mentioned in the transcript?

The given circumference of the second circle is 1600 pi.

How is the circumference related to the diameter?

The circumference is equal to pi times the diameter (C = πd).

What equation is used to find the diameter when the circumference is known?

The equation used is C = πd. By substituting the known circumference, we can solve for the diameter.

How is the diameter calculated from the given circumference of 1600 pi?

By setting 1600 pi equal to πd and dividing both sides by π, the diameter is found to be 1600 units.

Why is the number pi significant in the context of circles?

The number pi is significant because it is the ratio between the circumference and the diameter of a circle, a fundamental constant in mathematics.

Outlines

00:00

🧠 Understanding the Radius from Circumference

The video explains how to determine the radius of a circle given its circumference of 49π. Starting with a brief visual representation of a circle, it defines the circumference as 2πr, with π being the ratio of the circumference to the diameter. By setting 49π equal to 2πr, and dividing both sides by 2π, it simplifies to find that the radius r is 24.5 units.

🔍 Finding the Diameter from Circumference

The script transitions to another problem, calculating the diameter of a circle with a circumference of 1600π. By reiterating that the circumference can be expressed as π times the diameter, it shows that dividing 1600π by π gives a diameter of 1600 units. This reinforces the fundamental concept that π is the ratio of a circle's circumference to its diameter, highlighting the recurring significance of π in mathematics.

Keywords

💡Circumference

Circumference refers to the total distance around a circle. In the video, it is used to describe the measurement of a circle's perimeter, which is crucial in determining the circle's radius. The script mentions that the circumference is given as 49 pi, and this information is used to solve for the radius.

💡Pi

Pi, denoted as π, is a mathematical constant representing the ratio of a circle's circumference to its diameter. In the script, pi is used in equations to calculate the circumference and diameter of a circle, emphasizing its fundamental role in geometry.

💡Radius

The radius of a circle is the distance from its center to any point on its perimeter. The video script uses the concept of radius to derive the relationship between the circumference and the radius, ultimately solving for the radius when the circumference is given.

💡Diameter

Diameter is the longest distance across a circle, passing through its center. In the video, the diameter is related to the circumference through the constant pi, and the script uses this relationship to find the diameter when the circumference is known.

💡Ratio

A ratio is a comparison of two quantities, often expressed as a fraction. The script explains that pi is the ratio of the circumference to the diameter of a circle, highlighting the proportional relationship between these two measurements.

💡Hand-drawn Circle

A hand-drawn circle is a simple visual representation of a circle, typically not perfect. In the script, the instructor uses a hand-drawn circle to help visualize the concept of circumference and radius, making the mathematical concepts more tangible.

💡Visualization

Visualization is the process of creating a mental image or diagram to understand a concept better. The video script encourages viewers to visualize the circle and its properties, such as radius and circumference, to aid in the comprehension of the mathematical relationships.

💡Units

Units refer to the standard of measurement used to quantify physical properties. The script mentions units in the context of the circumference and diameter, indicating that the numerical values are dependent on the chosen unit of measurement.

💡Equation

An equation is a mathematical statement that asserts the equality of two expressions. The video script uses equations to express the relationship between the circumference, radius, and diameter, solving for one variable when the others are known.

💡Solve

To solve in mathematics means to find the value of an unknown variable in an equation. The script demonstrates how to solve for the radius and diameter of a circle by manipulating equations involving the circumference.

💡Fundamental

Fundamental refers to something that is basic or essential. The script describes circles and the concept of pi as fundamental in mathematics, indicating their importance and widespread application in various mathematical contexts.

Highlights

The circumference of a circle is given as 49 pi.

Encourages viewers to pause and solve the problem themselves.

Visualizes the problem by drawing a circle.

Explains that the circumference is two pi times the radius (2πr).

Pi is defined as the ratio between the circumference and the diameter.

Diameter is twice the radius (2r).

Circumference can be expressed as pi times two r.

Ratio between circumference and diameter is pi.

Solves the problem by substituting 49 pi for the circumference.

Divides both sides by two pi to solve for the radius.

Radius is calculated to be 24.5 units.

Introduces a new problem with a circle having a circumference of 1600 pi.

Asks what the diameter of the circle is.

Relates circumference to pi times the diameter.

Circles are fundamental in the universe, and pi is a mystical number.

Solves for the diameter by dividing 1600 pi by pi.

Diameter is found to be 1600 units.

Transcripts

- [Voiceover] Let's say that we know that the circumference

of a circle is 49 pi.

Based on that, let's see if we can figure out

what the radius of that same circle is going to be.

And I encourage you,

and I'll write equals here.

And I encourage you to pause the video,

and see if you can figure it out on your own.

Let's just draw the circle to help visualize it.

I'll just do a hand-drawn circle,

clearly not a perfect circle right over here.

We know that if its radius is of length r,

that the circumference

is going to be two pi times r.

So, I could write the circumference

is equal to two pi times r.

In fact, the number pi,

the standard definition for it,

is just the ratio between the circumference and the diameter

of a circle.

Now, why is that?

Well, if the diameter here

is two r, right?

We have r and then have another r.

We see that the circumference is pi times two r,

or we can say that the ratio between the circumference

and the diameter,

which is the ratio between c and two r,

that's just going to be pi.

Anyway, I've gone on longer than I need to

just to solve this problem.

We can go to this original formula here,

saying the circumference is two pi times r,

and we can just substitute in 49 pi for the circumference.

So, we could say 49 pi

is going to be equal to two pi times the radius.

Now, let's see, we can divide both sides by two pi

to solve for r.

So, dividing both sides by two pi.

On the right-hand side, the two pis cancel out.

On the left-hand side, pi divided by pi cancels out.

49 divided by two is 24.5.

So, if the circumference is 49 pi whatever units,

then the radius is going to be 24.5

of those units.

Let's do one more of these.

Let's say that we have a circle

whose circumference,

I'll just say C, is equal to 1600 pi.

My question is what is the diameter?

The diameter of the circle is equal to what?

Just as we said that the circumference

could be written as two pi r

or as pi times two r,

two r is just the diameter.

So, we could say that the circumference is equal to

pi times the diameter.

Once again, that comes out of that traditional definition

of pi as the ratio between the circumference

and the diameter.

You could say that the ratio between the circumference

and the diameter is equal to pi.

Circles are this very fundamental thing in the universe,

and you take the ratio

of the circumference and the diameter,

you get this magical and mystical number that we see

that keeps popping up in mathematics.

Anyway, back to the problem.

If we say the circumference is 1600 pi,

and this is equal to pi times the diameter,

we can just divide both sides by pi

to get the diameter,

which is going to be 1600.

The circumference is 1600 pi units,

whatever units those are, maybe meters.

Then, the diameter is just going to be 1600

of those units, or in this case, maybe meters.

And we're all done.