What part of a circle fits the definition below the distance along part of the circumference?

A circle is all points in the same plane that lie at an equal distance from a center point. The circle is only composed of the points on the border. You could think of a circle as a hula hoop. It's only the points on the border that are the circle. The points within the hula hoop are not part of the circle and are called interior points.

The distance between the midpoint and the circle border is called the radius. A line segment that has the endpoints on the circle and passes through the midpoint is called the diameter. The diameter is twice the size of the radius. A line segment that has its endpoints on the circular border but does not pass through the midpoint is called a chord.

The distance around the circle is called the circumference, C, and could be determined either by using the radius, r, or the diameter, d:

$$C=2\pi r$$

$$C=\pi d$$

A circle is the same as 360°. You can divide a circle into smaller portions. A part of a circle is called an arc and an arc is named according to its angle. Arcs are divided into minor arcs (0° < v < 180°), major arcs (180° < v < 360°) and semicircles (v = 180°).

The length of an arc, l, is determined by plugging the degree measure of the Arc, v, and the circumference of the whole circle, C, into the following formula:

$$l=C\cdot \frac{v}{360}$$

When diameters intersect at the central of the circle they form central angles. Like when you cut a cake you begin your pieces in the middle.

Example

As in the cake above we divide our circle into 8 pieces with the same angle. The circumference of the circle is 20 length units. Determine the length of the arc of each piece.

First we need to find the angle for each piece, since we know that a full circle is 360° we can easily tell that each piece has an angle of 360/8=45°. We plug these values into our formula for the length of arcs:

$$l=C\cdot \frac{v}{360}$$

$$l=20\cdot \frac{45}{360}=2.5$$

Hence the length of our arcs are 2.5 length units. We could even easier have told this by simply diving the circumference by the number of same size pieces: 20/8=2.5

Video lesson

What's the angle of the circle arc if we divide a cicle in $8$ equally sized pieces

A circle is the set of all points in the plane that are the same distance away from a specific point, called the center. The center of the circle below is point A. We call this circle “circle A,” and it is labeled $\bigodot A$.

Radius: The distance from the center of the circle to its outer rim.

Chord: A line segment whose endpoints are on a circle.

Diameter: A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.

Secant: A line that intersects a circle in two points.

Tangent: A line that intersects a circle in exactly one point.

Point of Tangency: The point where a tangent line touches the circle.

Figure $\PageIndex{2}$

The tangent ray $\overrightarrow{TP}$ and tangent segment $\overline{TP}$ are also called tangents.

Tangent Circles: Two or more circles that intersect at one point.

Concentric Circles: Two or more circles that have the same center, but different radii.

Congruent Circles: Two or more circles with the same radius, but different center

Figure $\PageIndex{3}$

What if you drew a line through a circle from one side to the other that does not pass through the center? What if you drew a line outside a circle that touched the circle at one point? What would you call these lines you drew?

Example $\PageIndex{1}$

Find the parts of $\bigodot A$ that best fit each description.

Figure $\PageIndex{4}$

A radius
A chord
A tangent line
A point of tangency
A diameter
A secant

Solution

\overline{HA}\) or \overline{AF}\)
$\overline{CD}$, $\overline{HF}$, or \overline{DG}\)
$\overleftrightarrow{BJ}$
$Point H$
$\overline{HF}$
$\overleftrightarrow{BD}$

Example $\PageIndex{2}$

Draw an example of how two circles can intersect with no, one and two points of intersection. You will make three separate drawings.

Solution

Figure $\PageIndex{5}$

Example $\PageIndex{3}$

Determine if any of the following circles are congruent.

Figure $\PageIndex{6}$

Solution

From each center, count the units to the outer rim of the circle. It is easiest to count vertically or horizontally. Doing this, we have:

$\begin{aligned} \text{Radius of } \bigodot A&=3\text{ unit } \\ \text{Radius of } \bigodot B&=4\text{ unit }\\ \text{Radius of } \bigodot C&=3\text{ unit }\end{aligned} $

From these measurements, we see that $\bigodot A\cong \bigodot C$.

Notice the circles are congruent. The lengths of the radii are equal.

Example $\PageIndex{4}$

Is it possible to have a line that intersects a circle three times? If so, draw one. If not, explain.

Solution

It is not possible. By definition, all lines are straight. The maximum number of times a line can intersect a circle is twice.

Example $\PageIndex{5}$

Are all circles similar?

Solution

Yes. All circles are the same shape, but not necessarily the same size, so they are similar.

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So far, we have discussed about the triangle and quadrilateral that have linear boundaries. Circle is a closed figure that has a curvilinear boundary.

When we think of circles, the very first thing that comes to our mind is its round shape, for example, bangles, coins, rings, plates, chapattis, pizzas, CDs etc. Wheels of a car, bus, cycle, truck, train, and aeroplane are also round in shape. If we take a stone, tie it to one end of a string and swing it in the air by holding the other end of the string, the path traced by the stone will be a circular path and it will make a circle.

Read More:

Circle: A circle is a collection of all those points in a plane that are at a given constant distance from a given fixed point in the plane.
Centre: Circle is a closed figure made up of points in a plane that are at the same distance from a fixed point, called the centre of the circle. In the figure O is the centre.

Radius: The constant distance from its centre is called the radius of the circle. In the figure, OA is radius
Chord: A line segment joining two points on a circle is called a chord of the circle. In the figure, AB is a chord of the circle. If a chord passes through centre then it is longest chord. PQ, PR, and ST are chords of the circle. Chord ST passes through the centre, hence it is a diameter.
Diameter: A chord passing through the centre of a circle is called the diameter of the circle. A circle has an infinite number of diameters. CD is the diameter of the circle as shown in the figure. If d is the diameter of the circle then d = 2r. where r is the radius. or the longest chord is called diameter. In the figure, AB is the diameter and the arcs CD and DC are semicircles.
Arc: A continuous piece of a circle is called an arc. Let A,B,C,D,E,F be the points on the circle. The circle is divided into different pieces. Then, the pieces AB, BC, CD, DE, EF etc. are all arcs of the circle.

Let P,Q be two points on the circle. These P, Q divide the circle into two parts. Each part is an arc. These arcs are denoted in anti-clockwise direction
Circumference of a circle: The perimeter of a circle is called its circumference. The circumference of a circle of radius r is 2πr.
Semicircle: The diameter of a circle divides the circle into two equal parts. Each part is called a semi-circle. We can also say that half of a circle is called a semi¬circle. In the figure, AXB and AYB represents two semi-circles.
Segment: Let AB be a chord of the circle. Then, AB divides the region enclosed by the circle (i.e., the circular disc) into two parts. Each of the parts is called a segment of the circle. The segment, containing the minor arc is called minor segment and the segment, containing the major arc, is called the major segment and segment of a circle is the region between an arc and chord of the circle.
Central Angles: Consider a circle. The angle subtended by an arc at the centre O is called the central angle. The vertex of the central angle is always at the centre O.
Degree measure of an arc: Degree measure of a minor arc is the measure of the central angle subtended by the arc.

The degree measure of the circumference of the circle is always 360°.
Interior and Exterior of Circle A circle divides the plane on which lies into three parts. (i) Inside the circle. which is called the interior of the circle (ii) Circle (iii) Outside the circle, which is called the exterior of the circle. The circle and its interior make up the circular region.
Sector: A sector is that region of a circular disc which lies between an arc and the two radii joining the extremities of the arc and the centre. OAB is a sector as shown in the figure. Quadrant: One fourth of a circular disc is called a quadrant.
Position of a point:
Point Inside the circle: A point P, such that OP < r, is said to lie inside the circle.

The point inside the circle is also called interior point. (Example : Centre of cirle)
Point outside the circle: A point Q, such that OQ > r, is said to lie outside the circle C (O, r) = {X, OX = r}
The point outside the circle is also called exterior point.
Point on the circle: A point S, such that OS = r is said to lie on the circle C(O, r) = {X ,OX = r}.
Circular Disc: It is defined as a set of interior points and points on the circle. In set notation, it is written as : C(O, r) = {X : P OX ≤ r}
Concentric Circles: Circles having the same centre and different radius are said to be concentric circles.
Remark. The word ‘radius’ is used for a line segment joining the centre to any point on the circle and also for its length.
Congruence of Circles & Arcs
Congruent circles: Two circles are said to be congruent if and only if, one of them can be superposed on the other, so as the cover it exactly. It means two circles are congruent if and only if, their radii are equal. i.e., C (O, r) and C (O’ , r) are congruent if only if r = s.

Congruent arcs: Two arcs of a circle are congruent, if either of them can be superposed on the other, so as to cover it exactly. It is only possible, if degree measure of two arcs are the same.

Example 1: Take two points A and B on a plane sheet. Draw a circle with A as a centre, AC as radius and B in its exterior.
Solution: Mark two points A and B on a paper. A • • B As the point B should be in the exterior of the circle, take A as the centre and radius (r) less than AB to draw a circle.

Example 2 :Find the diameter of the circle of radius 6 cm.
Solution: We know, Diameter = 2 × radius

∴ Diameter =2 × 6 cm =12 cm