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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 points that are equidistant from a given point. The given point is usually called the center of the circle. The distance between the center and any point on the circle is called a radius.

Radius circle.svg
Concept

Circumference of a Circle

The circumference is the distance around a circle, otherwise known as the circle's perimeter.
Animation unrolling a circle
The circumference is calculated by multiplying the diameter of the circle by

Since the diameter is twice the radius, it can also be calculated by multiplying by

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 $\Arc{AB}.$

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 $\Arc{ADB}.$ 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.
Rule

Arc Measure

The measure of an arc is equal to the measure of the central angle that creates it. Suppose arc is created when the central angle is drawn inside circle In the diagram above, the measure of arc

Suppose the measure of arc is given as The measure of the corresponding major arc, 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 $\Arc{AB}$ equals while the length of $\Arc{AB}$ 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.
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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 measures the space inside the circle.

For a circle with the radius the area can be calculated using the following formula.

Proof

Start by dividing a circle with radius into a number of equally sized sectors. Then distinguish the top and bottom half of the circle by filling them with different colors. Because the circumference of a circle is the arc length of each semicircle is half of it,

Rules Area of a Circle 2.svg

Now, imagine that all of the sectors of the circle are unfold. By placing the blue part as teeth pointing downward and the green part 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 is not easy to calculate the area of this figure, but note that if the circle is divided into more, even smaller sectors, then the figure will begin to look like a rectangle more and more.

The vertical sides becomes more vertical, and the horizontal sides becomes more horizontal. If the circle is divided into infinitely small sectors, the figure will become a perfect rectangle, with base and height Since the area of a rectangle is the product of its height and its base, the area of the figure created by the sectors of circle can be found as follows. Therefore, the area of the circle can also be found by this formula.

Concept

Sector of a Circle

A sector of a circle is a portion of the circle enclosed by two radii and an arc.

In the diagram, sector is created by and arc
Rule

Area of a Sector of a Circle

The area of a sector of a circle can be determined using the proportional relationship between the circle's and the sector's areas and measures. Substituting the known values gives the following. Therefore, the area of a sector of a circle can be found 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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