Geometry Segments in a Circle

Welcome back to Educator.com.0000

For this next unit, we are going to go over circles, and this first section of the unit is parts of the circle.0003

First, let's go over the different segments in a circle.0012

Now, here, right in the middle, that point right there is called the center--that is the center of the circle.0017

Now, whatever that point is...this point is labeled P; that means that this whole circle is going to be called circle P.0027

You are going to name the circle based on that center point.0035

And when you write it, it would be circle P; and you can also write it like this--a circle with a dot in the middle, P.0040

And that would mean circle P, just like we have triangle ABC and so on; this would be circle P.0053

Now, this first one right here: when you have a segment whose endpoints are on the circle, it is called a chord.0061

This right here is called a chord, and the endpoints must be on the circle.0074

Any time you have a segment with both endpoints on the circle, it is called a chord.0086

Now, this segment right here also has endpoints on the circle, so this segment is also known as a chord.0093

But more specifically, since it is passing through the center, this is called (I am sure you know) the diameter.0102

A diameter is still a chord, because any time you have a segment with endpoints on the circle, it is a chord.0113

But since this one is a little special, because it is passing through the center, we call it a diameter.0120

Now, what do we know as half the diameter? This right there is called the radius.0127

And then, when you have a line, this doesn't have endpoints; this is continuously going, and it is just passing through the circle at two points.0137

So, it is not a chord; now, if I just name this from this point to this point, then it could be called a chord, because I am calling it by its endpoints.0149

But this line itself, when it is just passing through the circle at two points--this is called a secant.0160

And again, it has to be passing through; it is different from a chord--a chord stops on one point of the circle and another point of the circle.0174

This doesn't stop; it just passes through, and it has to be passing through at two points--this part right here and this part right here.0183

Now, when you have either a line segment or a line passing through, but touching the circle in one point--0192

so it is like the secant; this is called a tangent, when it is touching at one point.0202

It could be a line segment or a line, but it just has to be touching the circle at one point; that is called a tangent.0212

These are the segments in a circle: chord, diameter, secant, and tangent--and radius, of course.0220

Next, we are going to go over circumference: if a circle has a circumference of C0239

(so then, C stands for circumference) and a radius of r units0244

(that means the r stands for radius), then C = 2πr, 2 times π times the radius; that equals circumference.0248

Now, what is circumference? If you take the circle right here from this point, and we are going to measure all the way around here, that is circumference.0256

Now, if I have a square, and I do the same thing--I start here, and I just measure all around the square,0271

or the rectangle, that would be called the perimeter.0282

A circumference and a perimeter are the same exact thing, except that perimeter is for anything with sides, and circumference is for the circle.0289

So, it is the perimeter of a circle; but we call it circumference, because there are no sides.0301

When you are given the radius--let's say that the radius is 5--then you are going to take the circumference,0310

and make it equal to 2 times the radius of 5, times π.0317

So then, the circumference would be 10π; and π, we know, is 3.14.0323

Well, it is more than that; it is actually an irrational number, so it goes on forever; but you can just use 3.14.0332

10 times 3.14 would be 31.4; that would be the circumference.0342

Now, you can also use a slightly different formula.0348

We know that, since 2 times π times r are all multiplied together, 2 times r is the diameter.0354

So, if I have 2, a radius right here to the center, and then from the center all the way to here, that is called the diameter.0362

So, you are going to multiply 2 times this number anyway, and that would be the diameter.0370

We can also say...instead of writing the 2 and the r, we can write d for diameter, and then π.0376

Circumference equals 2πr, or circumference equals diameter times π, dπ or πd.0387

Here, we have a right triangle within the circle, and we want to find the circumference of the circle.0403

To find the circumference, we need either (since this is the formula) 2πr, or the 2 and the r make up d, so dπ.0412

We need either the radius or the diameter.0423

Now, from this right triangle, the hypotenuse is the diameter of the circle.0427

So, as long as I can find the hypotenuse, then that would be my d.0433

How would you find the third side of this triangle?0440

Well, since it is a right triangle, I can use the Pythagorean theorem.0443

c² (that is the hypotenuse) = 12² + 5²; c² = 144 + 25.0450

So, c² = 169, which makes c 13; so this hypotenuse has a measure of 13.0468

And again, since this whole thing is the diameter, and the hypotenuse and the diameter are the same thing,0479

I can make that d; so this is also equal to diameter.0487

And instead of having to divide that by 2, and then having to multiply it again by 2 and multiply by π, remember: I can just use this.0494

Diameter is 13 times π; remember: for π, you can just use 3.14, even though it is an irrational number (it goes on forever).0505

13 times 3.14 would be 40.84; and you can just use your calculator for that.0518

You would just do 13 times...and your calculator might actually have a button that says π;0528

so instead of punching in 3.14, you can push the π button; so times π, equals, and then 40.84 is your answer.0535

We are just going to go through our examples now.0552

Use a circle to answer the following: now, this circle is labeled circle C.0556

Now, don't get this confused with circumference; it is not the same C--this is just the name of the point, C,0566

and circumference is capital C; that stands for circumference.0573

When I say circle C, it is just the name of the circle.0577

Name a chord of the circle: now, I know that a chord is when a segment has endpoints on the circle.0584

I can say that AB is a chord, because the endpoints are right here; and DE is also a chord,0596

even though DE is a diameter; as long as it has two endpoints on the circle, it is a chord.0603

So, it is a diameter, but it is also known as a chord; it is just passing through the center.0611

I am going to say AB, and I can say DE.0616

Now, we know that CF is not a chord, because we only have one endpoint on the circle; the other endpoint is at the center, so this is not a chord.0624

If DE right here, the diameter, is 10, what is CF?0635

CF is not a chord; it is a radius; CF is a radius (a radius, we know, is when we have one endpoint on the circle, and then the other at the center).0642

It is a radius; and all of the radii of a circle (plural is radii) are congruent.0655

So, if CF has a measure of 6, then this radius right here, CG--that also has a measure of 6.0668

If DE is 10, then I know that CD, which is half the diameter, which is a radius, is 5.0678

That means that CF, since it is also a radius, would also be 5; so CF = 5.0686

Is CE a chord? We know that a chord has two endpoints on the circle; this one does not, so this would be "no."0695

Why?--because CE has only one endpoint on the circle, and the other at the center, which would make it a radius.0708

I am going to write that over: CE has an endpoint right there on the circle, and another on the center; so that would make that a radius, not a chord.0755

The next example: Find the missing measures--so they are going to give us either the radius,0773

the diameter, or the circumference, and then we have to find the other two.0778

If the radius is 6, we know that the diameter is 2 times the radius; so if the radius is 6, then the diameter is 12.0782

And then, to find the circumference: the circumference is the diameter times π, so you can just punch in 12 times π, and that would be 37.7.0794

Then, the second one: the circumference is 43.96; so to find the radius and the diameter,0814

I am going to take the formula, and I am going to plug in the circumference.0822

So, 43.96 = dπ; now, I know that π is 3.14, but again, you can just use this on your calculator.0829

And since I am solving for d (this is my variable), I have to get rid of that π by dividing.0841

I can divide the π, and then your answer will be 14, so the diameter is 14.0850

Then you just divide that by 2 again, and then you are going to get the radius, which is 7.0863

Now, the diameter for this one is 2x; we know that the diameter is 2 times the length of the radius.0871

So, to find the radius, I just have to take the diameter and divide it by 2.0879

So, if you take 2x, and you are going to divide by 2, you are going to get x.0886

And then, to find the circumference, the circumference is going to be dπ; it is going to be 2x times π.0896

And you can either leave it like that, or you can just multiply the 2 and the π together: that will be 2 times x times 3.14, which would make that 6.28x.0909

And that would just be your answer; when you solve for C, that is your answer, 6.28x.0935

Or it could be just 2x times π.0945

The next example: the circumference is 37.68; find the perimeter of the triangle.0953

We have a triangle that is inscribed; that just means that it is inside the circle, with all of the endpoints on the circle.0960

And it is a right triangle; they give us the circumference, and we want to find the perimeter of the triangle.0972

Since it is a triangle, and it has sides, we call it "perimeter" instead of "circumference."0982

The circumference...we are going to use π times d, because that would be the hypotenuse of this triangle.0986

And then, we are going to plug this in the formula, so that we can solve for d; and that will give us the hypotenuse.0997

37.68 = πd; then I can divide the π, and d is going to be 12.1007

I have a calculator here on my screen; you are just going to take 37.68, and divide it by...1023

and there is a π button here, so...it is going to be 12.1032

Then, I am going to label that onto this triangle; that is 12.1041

I have the hypotenuse, and I have one side, 8.1048

In order for me to find the perimeter, I need all of the sides--I need the measures of all three sides.1051

So, here, this side is missing; how would I find that side?1056

If it is a right triangle, and you have the measures of two sides, then you would use the Pythagorean theorem.1061

12², the hypotenuse squared, equals 8² + x².1072

144 = 64 + x²; subtract 64; you are going to get 80 = x².1080

From there, take the square root; you are going to get 8.9, so this right here is 8.9.1098

And now, to find the perimeter, because that is what they are asking for: the perimeter of the triangle would be 12 + 8 + 8.9.1117

This is 20 + 8.9; that will be 28.9, so that is your answer.1128

Now, if you were to have units here--if this were, let's say, inches--then I know that my perimeter is just going to be in inches.1139

And so, also, for circumference, it is just the units; it is not units squared.1151

We are not looking for the area of anything--circumference and area; it is just that we keep the same units.1157

For the fourth example, we are going to find the circumference of each circle.1169

Now, in this one, we have a square, and we have an inscribed circle.1174

And "inscribed," again, means when you have something--a shape--inside of another shape, and all of the endpoints are touching.1180

Here, this is all touching here; this is circle P; the side of the square is 10; and I have to find the circumference.1191

Now, I know that for circumference, I need either radius or the diameter.1207

Let's say diameter: now, if I draw the diameter like this, I know that this diameter is going to be the same measure as this side,1214

because if this is the endpoint from here to here, that is the same thing as this endpoint from here to here.1227

And this is 10, so then, this diameter would also be 10; so the circumference is going to be 10 times 3.14, which means it is going to be 31.4.1232

And then, the next one: we have an inscribed triangle; endpoints are all on the circle.1252

This is an isosceles triangle; if this is 4, then this also has to be 4.1264

So again, to find the circumference, I need the diameter, or I need the radius.1270

The diameter is also the hypotenuse; so I am going to use the Pythagorean theorem to look for this measure right there.1276

4...let's see, c² = 4² + 4²; c² = 16 + 16, or 32...√32 = 5.66.1284

And instead of labeling that c, I will actually label that as d, because it is the hypotenuse, but it is also the diameter.1318

Its diameter is that; and then, the circumference is going to be, again, π times d; so that would be 5.66 times π, or 3.14.1330

And your circumference is 17.77.1350

That is it for this lesson; thank you for watching Educator.com.1360