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Circles are some of the most important geometric shapes we work with. You've probably learned how to calculate the area or circumference of a circle in your past math classes, but there's so much more that can be done with them. In this section, we'll ground the basics of circle geometry in more rigorous more mathematical ideas.
We'll start by reviewing the fundamentals to get comfortable working with circle vocabulary.
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Circumference & Area of a circle
The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called π (pi). Since the diameter is twice the radius, the circumference of a circle is
The area of a circle is
where r is the radius of the circle.
Checkpoint
A circle has area 8π.
Find the circumference of the circle.
Select the correct option
The formula for the circumference of a circle C=2πr only provides a way to calculate the length of the entire boundary of a circle, but there are many cases where we want to know the length of a smaller part of a circle's boundary.
Discussion
Below is a circle of radius r. You can drag the point on the circle's exterior circle around to increase or decrease the highlighted portion of circumference. The angle between the "starting point" (at the end of the segment marked r) and the draggable point, θ, will update in real time as you drag it further around.
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The circumference C of the entire circle, as given by the formula, is 2πr. Try dragging the highlighted point around the circle and considering the segment's length at different values of θ.
What is the length of the purple portion when
θ=180°?
θ=90°?
θ=60°?
Can you come up with a general formula, in terms of r, for the length of the purple section that depends on the value of the interior angle θ?
To see why the purple section always represents the same fraction of the full circle as θ is of 360°, start with the formula for the circumference of a circle:
When θ=180°:
180° is half of 360°, so the purple section is half the circumference:
When θ=90°:
90° is one quarter of 360°, so the purple section is one quarter of the circumference:
When θ=60°:
60° fits into 360° six times, so it is one sixth of the circle:
In each case, the length of the purple section is the same fraction of the circumference as the angle is of 360°.
For a general angle θ (in degrees), the length is 360θ of the full circle's circumference, so:
In fact this formula works for any interior angle θ. We call the purple highlighted section an arc.
Arc length (degrees)
An arc is part of the circumference of a circle. It is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
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Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, the arc length is calculated as
Checkpoint
A circle of radius 2 has an arc with length 2π.
Find the value of the internal angle θ.
Select the correct option
Just as an arc is a portion of a circle's circumference, we call a "pizza slice" portion of a circle's area a sector.
Discussion
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Can you come up with a formula for the area of a sector that, similar to the one for arc length, depends on the radius r and internal angle θ?
The key idea is that a sector is just a fraction of the whole circle, the same fraction that its arc length is of the full circumference.
We know
and for an internal angle θ in degrees the arc length is
Since the sector “cuts out” exactly the same fraction 360∘θ of the circle’s area, and the full area is πr2, its area A is
That is the desired formula.
Sector (degrees)
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is πr2, and there are 360 degrees of rotation in a circle. Therefore, a sector with central angle θ is 360°θ of a full circle, and has area
Checkpoint
A sector with central angle θ=40° has area A=4π.
Find the length of the circle's radius r.
Select the correct option
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Up to this point, the circle measurements we've discussed have relied entirely on the value of an interior angle, θ, which we measure in degrees. This works perfectly well, but as you may have noticed, sometimes results in numbers that are difficult to work with or annoying to calculate, and degrees aren't always the most intuitive measurement.
We can also describe θ using units that are tied more naturally to circle geometry. Instead of assigning θ a value out of 360°, these units use the known fact that the circumference of a circle is 2πr and "work backward" to describe what portion of the total circumference an arc or sector represents.
We call these units radians.
Definition of a radian
One radian is the interior angle of an arc which has a length equivalent to the radius r of the circle. Since the circumference of a circle is given by 2πr, then, there are 2π total radians in a circle (the equivalent of 360°).
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Discussion
Given that there are 360 degrees and 2πr radians in a circle, can you come up with a way to convert from radians to degrees and vice versa?
We know that a full circle measures 360∘ or 2π radians. Hence
and dividing by 360 gives
so to convert degrees to radians
and inverting, since 1 rad=180/π∘,
Converting Between Radians & Degrees
Since the perimeter of a full circle is 2πr, the angle θ corresponding to a full circle (360°) is
So
Some key angles in radians and degrees:
Example
Convert the angle 2rad to degrees:
so
Checkpoint
Convert the following angles to degrees:
125π
127π
1211π
Select the correct option
An angle's value in radians is given by "how many radiuses" the corresponding arc . This gives a straightforward way of calculating arc length based on the value of θ in radians.
Arc length (radians)
An arc is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
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Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, the arc length is calculated as
Checkpoint
A circle has area A=49π.
Find the length l of an arc with interior angle θ=3πrad.
Select the correct option
We can also calculate sector area in terms of radians.
Sector area (radians)
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is πr2, and there are 2π radians in a circle. Therefore, a sector with central angle θ is 2πθ of a full circle, and has area
so
Exercise
Find the area of the sector below.
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