Radius & diameter from circumference | High School Geometry | High School Math | Khan Academy
Summary
TLDRThis 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
🧠 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.
Mindmap
Keywords
💡Circumference
💡Pi
💡Radius
💡Diameter
💡Ratio
💡Hand-drawn Circle
💡Visualization
💡Units
💡Equation
💡Solve
💡Fundamental
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.
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