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Circles

Using Circumference and Area

Circumference and area are two measurements that can be used to analyze circles. The first measures the boundary or perimeter of a circle, while the second measures the space inside a circle. Sometimes, only a portion of the circumference and/or area — arcs and sectors, respectively — are explored.

Concept

Circle

A circle is the set of all the points in a plane that are equidistant from a given point. There are a few particularly notable features of a circle.

  • Center - The given point from which all points of the circle are equidistant. Circles are often named by their center point.
  • Radius - A segment that connects the center and any point on the circle. Its length is usually represented algebraically by
  • Diameter - A segment whose endpoints are on the circle and passes through the center. Its length is usually represented algebraically by
  • Circumference - The perimeter of a circle, usually represented algebraically by
In the applet, the center is labeled Therefore, the circle can be referred to as or circle
parts of a circle
In any given circle, the lengths of any radius and any diameter are constant. They are called the radius and the diameter of the circle, respectively.

Rule

Circumference of a Circle

The circumference of a circle is calculated by multiplying its diameter by

This can be visualized in the following diagram.
Animation unrolling a circle

Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying by

Proof

Consider two circles and their respective diameters and circumferences.

two circles and their diameters
By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional. This proportion can be rearranged.
Rearrange equation
Therefore, for any two circles, the ratio of the circumference to the diameter is always the same. This means that this ratio is constant. This constant is defined as With this information, it can be shown that the circumference of a circle is the product between its diameter and

Concept

Central Angle

An angle whose vertex lies at the center of a circle and whose legs are radii is called a central angle.

In this diagram, is a central angle of the circle. Note that the measure of the intercepted arc of a central angle is equal to the measure of the central angle.

Why


Recall that both a circle and a complete angle measure degrees. Since the vertex of a central angle lies at the center of a circle, both the central angle and its intercepted arc represent the same portion of the circle. Use the slider in the applet to see this relationship.

Therefore, the central angle and its intercepted arc have the same measure.

Concept

Minor and Major Arc

Suppose is drawn in circle such that it is a central angle. If is less than the endpoints of the angle, and create a minor arc on the circle. This arc can be named

If is greater than the endpoints of the angle create a major arc on the circle. In other words, the points on the circle not included in the minor arc create a major arc.

A major arc can be named using its endpoints and another point on the arc. Therefore, the major arc can be named In the diagram above,

  • is the center of the circle,
  • is a central angle of the circle,
  • arc is a minor arc, and
  • arc is a major arc.

Concept

Arc Measure

The measure of an arc is equal to the measure of its corresponding central angle. In the diagram, the measure of is equal to the measure of

Suppose the measure of is given as The measure of the corresponding major arc can be found by subtracting from the measure of the entire circle.

Concept

Arc Length

An arc length is a portion of the circumference of a circle.

In the diagram, the measure of equals while the length of corresponds to the portion of the circumference the red arc spans. To find it, the relationship between arc length and arc measure can be used.
fullscreen
Exercise

Find the measure of arcs and

Show Solution
Solution
To begin, let's make sense of the given information. As can be seen, is the length of arc and is the radius of the circle. Because arc length is given and arc measure needs to be determined, we can use the relationship between the two. To find the measure of arc we must first calculate the circumference of the circle. Now, we can substitute the known values for circumference and arc length into the equation to solve for the measure of arc
Thus, the measure of arc is approximately Notice that arc is a minor arc and arc is its corresponding major arc. As a result, we can subtract from to find the measure of arc Thus, the measure of arc is approximately
fullscreen
Exercise

Find the length of arc

Show Solution
Solution

To begin, we can make sense of the given information. The circle has a radius of inches and the arc mesaures The relationship between arc measure and arc length can be used. First, the circumference of circle can be calculated as follows.

The length of arc can be calculated by substituting the values for circumference and arc measure, and solving.
Thus, the length of arc is approximately

Concept

Radian Measure

In the diagram below, circle has a radius of unit, and circle has a radius of units.

The measures of arcs and are equal because they are created by the same central angle. Because circle has a radius of the length of arc can be found using the definition of arc length. Because all circles are similar, circle circle Thus, the ratios of corresponding parts between the circles are proportional. An equation relating arc length to radius can be written. Simplifying the right-hand side and substituting the found value for the length of arc gives the following. The equation above shows that the length of arc is proportional to the radius of the circle. In fact, the expression is called the constant of proportionality, and defines the radian measure of the central angle associated with the arc. Because the circumference of a circle with radius is the radian measure of a circle is radians — which corresponds to

Rule

Translating between Radians and Degrees

Because angles can be measured in both radians and degrees, it can be useful to translate between the two. The concept of radian measure asserts that one revolution around a circle corresponds to exactly radians. It is also known that this same distance can be expressed as Thus, Simplifying gives the following conversion factor.

This conversion factor can be used to translate from radians to degrees and vice versa.
fullscreen
Exercise

Convert to radians and to degrees.

Show Solution
Solution
To convert both measures, we can use the following relation. The following proportion can be written to convert to radians. Solving the equation for will give the corresponding value.
Thus, corresponds to approximately radians. In a similar way, we can convert radians to degrees. Thus, radians corresponds to

Rule

Area of a Circle

The area of a circle is the product of and the square of its radius.

Proof

Informal Justification

A circle with radius will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is the arc length of each semicircle is half this value,

Rules Area of a Circle 2.svg

Now, the above sectors will be unfolded. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. Therefore, the area of the figure below should be the same as the circle's area.

It can be noted that if the circle is divided into more and smaller sectors, then the figure will begin to look more and more like a rectangle.
Here, the shorter sides become more vertical and the longer sides become more horizontal. If the circle is divided into infinitely many sectors, the figure will become a perfect rectangle, with base and height Since the area of a rectangle is the product of its and its the following formula can be derived.

It was shown that the area of a circle is the product of and the square of its radius.

Concept

Sector of a Circle

A sector of a circle is a portion of the circle enclosed by two radii and their intercepted arc.
Sector of a Circle
In the diagram, sector is created by and

Rule

Area of a Sector of a Circle

The area of a sector of a circle is calculated by multiplying the circle's area by the ratio of the measure of the central angle to

From the fact that equals an equivalent formula can be written if the central angle is given in radians.

Since the measure of an arc is equal to the measure of its central angle, the arc measures Therefore, by substituting for another version of the formula is obtained which can also be written in degrees or radians.


or

Proof

Consider sector bounded by and

sector of a circle

Since a circle measures this sector represents of Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to

Recall that the area of a circle is By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained. Therefore, the area of a sector of a circle can be found by using the following formula.

fullscreen
Exercise

Find the area of the sector.

Show Solution
Solution
To begin, notice that the arc corresponding with the given sector measures Notice also that the radius of the circle is cm. We can substitute these values into the formula for the area of a sector and solve.
Thus, the area of the sector is approximately
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