Lesson Video: Circles | Nagwa Lesson Video: Circles | Nagwa

# Lesson Video: Circles Mathematics

In this video, we will learn how to use the properties of a circle and its parts in order to solve problems.

09:54

### Video Transcript

In this video, weโll learn how to identify the basic parts of the circle, such as the radius, chord, and diameter, and use their properties to solve problems. A circle, which is not a polygon since it has no straight edges, is a one-sided, two-dimensional shape. It will have a center which is usually defined by the letter ๐. And all points on the circle itself, thatโs this curve line, are equidistant from this center. In other words, the lines joining the center to these points are equal in length. Letโs begin by having a look at some of the other key parts of a circle.

Label each part of the circles. Then, we have a few terms. We have radius, diameter, circumference, chord, tangent, arc, sector, minor segment, and major segment.

Letโs go through each of these words term by term and give them a definition. We begin with the radius. The radius is any line which is made by joining the point at the center of our circle to a point on the circumference. If we look at the circle at the bottom of our diagram, we see that thereโs a radius marked here. In fact, thereโs also a radius in our top circle. Remember, itโs simply the line that joins the point at the center to any point on the circumference. For this reason, we can draw an infinite number of radii in our circles.

And then we move on to the diameter. The diameter is closely linked to the radius. This time, though, the diameter is found by joining points on opposite sides of the circle with a straight line. And this straight line must pass through the center. And so, if we look at our first circle, we see we have a diameter here. We then move on to the circumference. We say that the circumference is the perimeter of the circle. Itโs the outside. And so, we label the curve line that encloses our circle in the upper diagram as the circumference.

Next, we move on to the chord. A chord is any line that joins two points on the circumference of the circle. Now, notice weโve already labeled the chord on our first circle as a diameter. But on our second circle, we have a chord here. And what about the tangent? A tangent to a circle is a straight line which touches the circle at only one point. We only have one line. This is on the outside of our circles, and itโs this one. So, this is the tangent. The next word is the arc. Essentially, the arc of a circle is just part of its circumference. Thatโs this line here.

Next, we have the sector. The sector is the portion of the circle that looks a little bit like a piece of pie. Itโs this one here, and itโs the entire shape. Finally, we have a minor segment and a major segment. The minor segment is a part of the circle that looks a little bit like an orange segment. Itโs this slice here. Once again, we have the entire shape, and itโs made by portioning up the circle using a chord. It follows that the major segment is the rest of the circle. Itโs this one here.

We must be able to identify each of the parts of these circles by heart. And weโre next going to see how we can use some of the properties of these parts of the circles to solve problems.

Are all the radii of a circle equal in length?

Radii is just the plural of radius. And the radius of the circle is any line that joins the center of that circle to a point on its circumference. So, if we take a circle center ๐, we could have a radius here. It joins the center to a point on its circumference. We could have one here or one over here. Note that since the line begins at the center of the circle each time and ends on the circumference, this line must be the same length as this line and this line. And so, we say yes that all radii of a circle are equal in length.

And in fact, this is a really useful fact. It means we can solve problems involving angles in a circle by creating isosceles triangles with any two radii.

Is the diameter the longest chord in a circle?

Remember, a chord is a line that joins one point on the circumference of a circle to another. So, if we take a circle whose center is ๐, we could have a chord here, one here, or one here. There are an infinite number of chords that we can draw. Note that to create the longest chord, thatโs the longest line by joining two points on the circumference, weโre going to need to find the widest part of the circle.

Now, this line must pass through the center of the circle. But of course, we know that the diameter is the line that passes through the center of the circle and joins two points on its circumference. And this means that we can say that yes, the diameter is the longest chord in a circle.

Next, we look at the relationship between the radius and the diameter of a circle.

The length of the radius of a circle is what the length of its diameter.

Remember, the radius of a circle is the straight line that joins the point at the center to any point on the circumference of the circle. So, if we take a circle whose center is ๐, we could have a radius here. And what about the diameter? Well, the diameter also passes through the center of the circle. But this time, itโs a straight line that joins two points on the circumference. So, the diameter of this circle is this line here.

But of course, If we look carefully at the diameter, we can see that we can make up of one, two radii. And so, this means that a diameter of the circle must be twice or double the length of the radius. And in turn, we can say that the radius of the circle must therefore be half the length of its diameter.

Weโll now look at how to use these features to solve problems involving circles.

If the diameters of circles ๐ and ๐ are two centimeters and six centimeters, respectively, determine the length of the line segment ๐๐.

The line segment ๐๐ is shown. Itโs a single straight line. And if we look carefully, we see that each part of the straight line is made up by joining the point at the center of each circle to a point on its circumference. That means this first part of the line segment is the radius of ๐. The radius of a circle is the line that joins a point on the circumference to its center. Similarly, the second part of our line segment is the radius of our circle ๐. This means that the line segment ๐๐ is the sum of these. Itโs the length of the radius of ๐ plus the length of the radius of ๐.

Now, the problem is weโre not actually given information about the radii of our circles. We are, however, told that the diameters of our circles are two centimeters and six centimeters, respectively. Now, we know that the diameter of a circle is twice or double the length of the radius. And we can equivalently say that this means that the radius must be half the length of the diameter. Of course, to find half of a number, we divide it by two. So, this means the radius of ๐ is two divided by two, which is equal to one centimeter.

Similarly, we halve the length of the diameter of circle ๐. Thatโs six divided by two, which is three centimeters. So, ๐๐ is the sum of these two lengths. Itโs one plus three, which is equal to four or four centimeters. The length of line segment ๐๐ is four centimeters.

Weโll consider one more problem-solving example.

In the figure, ๐๐ด is equal to 34 centimeters. Determine the length of ๐ถ๐ธ.

Weโve been given part of a circle and some information about the length of the line ๐๐ด. Now, ๐๐ด is this line here. And letโs imagine that we actually had a full circle. If we do, we see that ๐๐ด is the radius. Itโs the line that joins the center to a point on the circumference of the circle. In fact, we have a radius here. This line here is also a radius. Now, weโre told that ๐๐ด, the radius of the circle, is 34 centimeters. So, each of these lines that weโve added must all be 34 centimeters.

Now, in fact, if we join ๐ to ๐ท to create the line segment ๐๐ท, we see that that is also the radius of the circle. So, ๐๐ท must also be 34 centimeters. Now, you might be thinking that you need to use something like Pythagorasโ theorem next. But no, we see if we look carefully that ๐๐ถ๐ท๐ธ is a rectangle. And we know that the diagonals of a rectangle are equal in length. This means that ๐ถ๐ธ, which is also a diagonal, must be equal in length to ๐๐ท. We established ๐๐ท was equal to 34 centimeters. So, this means that ๐ถ๐ธ is also equal to 34 centimeters.

In this video, weโve learned that the key parts of a circle are the radius, diameter, circumference, chord, tangent, arc, minor segment, major segment, and sector. We saw that all radii of a circle are equal in length, that the length of the radius is exactly half the length of the diameter, and that the diameter is the longest possible chord in a circle. We also saw that these features can help us to solve problems involving parts of a circle.

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