Circumference
Perimeter describes the distance around a figure. The perimeter of a circle has a different name; it is called circumference. For circles, it isn’t as simple as measuring each side length, since the distance around a circle is curved. Measuring the distance around a circle is tricky - it’s almost like you need to take a bendable ruler to measure it! |
But since all circles are similar, we merely need to know the length of the radius to find a circle’s circumference:
\(\text{Circumference = diameter}\cdot\pi\) = \(2\cdot\text{radius}\cdot\pi\)
\(C = d\pi = 2r\pi\)
Example 1:
If the diameter of a circle is \(12\) inches, what is the circumference of the circle? Leave your answer in terms \(\pi\) of and then round it to the nearest tenth.
Solution:
Using the circumference formula:
\(C = d\pi = 12\pi\) inches
\(C \approx 37.699111...\) inches
\(C \approx 37.7\) inches
Arc Length
Suppose you don’t want to find the entire length around the circle, just a part of it. We are describing arc length. Arc length is a fraction of the circle’s circumference. Before we can find arc length, we need to know how much of the entire circle the arc will take up.
To determine the fraction of the circle, recall that the measure of an entire circle is \(360^{\circ}\). Let’s use the following example to first determine the fraction of the circle:
Example 2:
Determine what fraction \(\overset{\Huge{\frown}}{AB}\) is of \(\bigodot C\).
a) b) c) d)
\(\text{Circumference = diameter}\cdot\pi\) = \(2\cdot\text{radius}\cdot\pi\)
\(C = d\pi = 2r\pi\)
Example 1:
If the diameter of a circle is \(12\) inches, what is the circumference of the circle? Leave your answer in terms \(\pi\) of and then round it to the nearest tenth.
Solution:
Using the circumference formula:
\(C = d\pi = 12\pi\) inches
\(C \approx 37.699111...\) inches
\(C \approx 37.7\) inches
Arc Length
Suppose you don’t want to find the entire length around the circle, just a part of it. We are describing arc length. Arc length is a fraction of the circle’s circumference. Before we can find arc length, we need to know how much of the entire circle the arc will take up.
To determine the fraction of the circle, recall that the measure of an entire circle is \(360^{\circ}\). Let’s use the following example to first determine the fraction of the circle:
Example 2:
Determine what fraction \(\overset{\Huge{\frown}}{AB}\) is of \(\bigodot C\).
a) b) c) d)
Solution:
a) Since there are \(360^{\circ}\) in a circle, we will make a fraction with \(m\overset{\Huge{\frown}}{AB}\) in the numerator and \(360^{\circ}\) in the denominator. Perhaps you may have noticed that \(\overset{\Huge{\frown}}{AB}\) is a semicircle, which measures \(180^{\circ}\). A semicircle is half of the circle, so the fraction we are looking for is \(\Large\frac{1}{2}\). Another way to look at it is: \(\Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\frac{180^{\circ}}{360^{\circ}}=\frac{1}{2}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up half of the entire circle!
b) Since the central angle measures \(90^{\circ}\), which means that \(m\overset{\Huge{\frown}}{AB}\) is also \(90^{\circ}\). So: \(\dfrac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\dfrac{90^{\circ}}{360^{\circ}}=\dfrac{1}{4}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up a quarter (or one fourth) of the entire circle.
c) Since the central angle measures \(120^{\circ}\), which means that \(m\overset{\Huge{\frown}}{AB}\) is also \(120^{\circ}\). So: \(\dfrac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\dfrac{120^{\circ}}{360^{\circ}}=\dfrac{1}{3}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up one third of the entire circle.
d) Since the central angle measures \(132^{\circ}\), which means that \(m\overset{\Large{\frown}}{AB}\) is also \(132^{\circ}\). So: \(\dfrac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\dfrac{132^{\circ}}{360^{\circ}}=\dfrac{11}{36}=\normalsize0.3\overline{6}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up about \(36.7%\) of the entire circle.
Let’s use what we found above to calculate the length of an arc!
a) Since there are \(360^{\circ}\) in a circle, we will make a fraction with \(m\overset{\Huge{\frown}}{AB}\) in the numerator and \(360^{\circ}\) in the denominator. Perhaps you may have noticed that \(\overset{\Huge{\frown}}{AB}\) is a semicircle, which measures \(180^{\circ}\). A semicircle is half of the circle, so the fraction we are looking for is \(\Large\frac{1}{2}\). Another way to look at it is: \(\Large\frac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\frac{180^{\circ}}{360^{\circ}}=\frac{1}{2}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up half of the entire circle!
b) Since the central angle measures \(90^{\circ}\), which means that \(m\overset{\Huge{\frown}}{AB}\) is also \(90^{\circ}\). So: \(\dfrac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\dfrac{90^{\circ}}{360^{\circ}}=\dfrac{1}{4}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up a quarter (or one fourth) of the entire circle.
c) Since the central angle measures \(120^{\circ}\), which means that \(m\overset{\Huge{\frown}}{AB}\) is also \(120^{\circ}\). So: \(\dfrac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\dfrac{120^{\circ}}{360^{\circ}}=\dfrac{1}{3}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up one third of the entire circle.
d) Since the central angle measures \(132^{\circ}\), which means that \(m\overset{\Large{\frown}}{AB}\) is also \(132^{\circ}\). So: \(\dfrac{m\overset{\Huge{\frown}}{AB}}{360^{\circ}}=\dfrac{132^{\circ}}{360^{\circ}}=\dfrac{11}{36}=\normalsize0.3\overline{6}\). This means that \(\overset{\Huge{\frown}}{AB}\) takes up about \(36.7%\) of the entire circle.
Let’s use what we found above to calculate the length of an arc!
Finding Arc Length Given Circumference
Example 3:
If the circumference of a circle is \(70\pi\) cm, what is the length of an arc that is one-tenth of the circumference?
Solution:
Let’s draw a picture of this situation.
Example 3:
If the circumference of a circle is \(70\pi\) cm, what is the length of an arc that is one-tenth of the circumference?
Solution:
Let’s draw a picture of this situation.
Suppose the pink arc below represents \(\dfrac{1}{10}\) of the entire circle.
Recall that arc length is a fraction of the circle’s circumference. Since we know that we are looking for \(\dfrac{1}{10}\) of the circumference, let’s calculate the following:
\(s = \dfrac{1}{10}(70\pi)\)
\(s = \dfrac{70\pi}{10}\)
\(s = 7\pi\) cm
So the length of that arc is \(7\pi\) cm
Finding Arc Length Given Radius
Your approach will differ slightly if you are given the length of the radius. Let’s try the example below.
Example 4:
Given \(\bigodot N\), find the length of \(\overset{\Huge{\frown}}{JE}\). Keep your answer in terms of \(\pi\).
\(s = \dfrac{1}{10}(70\pi)\)
\(s = \dfrac{70\pi}{10}\)
\(s = 7\pi\) cm
So the length of that arc is \(7\pi\) cm
Finding Arc Length Given Radius
Your approach will differ slightly if you are given the length of the radius. Let’s try the example below.
Example 4:
Given \(\bigodot N\), find the length of \(\overset{\Huge{\frown}}{JE}\). Keep your answer in terms of \(\pi\).
Solution:
Let’s use the formula for arc length to find it:
\(s=\dfrac{\text{arc measure}}{360^{\circ}}\cdot C\)
\(=\dfrac{\text{arc measure}}{360^{\circ}}\cdot2r\pi\)
\(=\dfrac{80^{\circ}}{360^{\circ}}\cdot2(3)\pi\)
\(=\dfrac{2}{9}\cdot(6\pi)\)
\(=\dfrac{12\pi}{9}\)
\(=\dfrac{4\pi}{3}\) units
Using Arc Length
Example 5:
Suppose in \(\bigodot L\) that \(m\overset{\Huge{\frown}}{ME}=60^{\circ}\) and the length of \(\overset{\Huge{\frown}}{ME}\) is \(10\) feet. What is the radius of \(\bigodot L\)? Round your answer to the nearest tenth of a foot.
Solution:
We will still use the arc length equation, but we will need to use order of operations to find the length of the radius:
\(s=\dfrac{\text{arc measure}}{360^{\circ}}\cdot2r\pi\)
\(10=\dfrac{60^{\circ}}{360^{\circ}}\cdot2r\pi\) Substitute the values that we know.
\(10=\dfrac{1}{6}\cdot2r\pi\) Simplify the fraction
\(60=2r\pi\)Use inverse operations to solve for \(r\)
\(\dfrac{60}{\pi}=\dfrac{2r\pi}{\pi}\)
\(19.0989859...= 2r\)
\(9.5492965...=r\)
\(r\approx9.5\) ft
We can also use arc length combined with what we already know about perimeters of other figures. We simply must break up the perimeter into straight segments and arcs. See the example below.
Example 6:
In the diagram below, points \(A, J,\) and \(M\)are centers of circles. Also, \(\overline{FN}\) and \(\overline{CD}\) are tangents. Find the perimeter of the following region bounded by circular arcs.
Let’s use the formula for arc length to find it:
\(s=\dfrac{\text{arc measure}}{360^{\circ}}\cdot C\)
\(=\dfrac{\text{arc measure}}{360^{\circ}}\cdot2r\pi\)
\(=\dfrac{80^{\circ}}{360^{\circ}}\cdot2(3)\pi\)
\(=\dfrac{2}{9}\cdot(6\pi)\)
\(=\dfrac{12\pi}{9}\)
\(=\dfrac{4\pi}{3}\) units
Using Arc Length
Example 5:
Suppose in \(\bigodot L\) that \(m\overset{\Huge{\frown}}{ME}=60^{\circ}\) and the length of \(\overset{\Huge{\frown}}{ME}\) is \(10\) feet. What is the radius of \(\bigodot L\)? Round your answer to the nearest tenth of a foot.
Solution:
We will still use the arc length equation, but we will need to use order of operations to find the length of the radius:
\(s=\dfrac{\text{arc measure}}{360^{\circ}}\cdot2r\pi\)
\(10=\dfrac{60^{\circ}}{360^{\circ}}\cdot2r\pi\) Substitute the values that we know.
\(10=\dfrac{1}{6}\cdot2r\pi\) Simplify the fraction
\(60=2r\pi\)Use inverse operations to solve for \(r\)
\(\dfrac{60}{\pi}=\dfrac{2r\pi}{\pi}\)
\(19.0989859...= 2r\)
\(9.5492965...=r\)
\(r\approx9.5\) ft
We can also use arc length combined with what we already know about perimeters of other figures. We simply must break up the perimeter into straight segments and arcs. See the example below.
Example 6:
In the diagram below, points \(A, J,\) and \(M\)are centers of circles. Also, \(\overline{FN}\) and \(\overline{CD}\) are tangents. Find the perimeter of the following region bounded by circular arcs.
Solution:
First, we will break the diagram into straight segments and circular arcs. We will use the arc length formula to find the length of the arcs, and then add together each length to get the total perimeter. Let’s watch this video for its solution!
First, we will break the diagram into straight segments and circular arcs. We will use the arc length formula to find the length of the arcs, and then add together each length to get the total perimeter. Let’s watch this video for its solution!
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