Suppose we don’t want to find the area of the entire circle, but just a part of it. This is where new vocabulary comes in: A sector is a section of a circle that is bounded by radii and a circular arc. We name sectors using the points on the circle and the radius. Take a look at some of the examples and non-examples below:
Example 1:
Given point \(A\) is the center of the circle, determine which of the following shaded regions is a sector and which is not.
a) b) c)
Example 1:
Given point \(A\) is the center of the circle, determine which of the following shaded regions is a sector and which is not.
a) b) c)
Solution:
a) Since the shaded region is bounded by two radii (\(\overline{AP}\) and \(\overline{AM}\)) and a circular arc (\(\overset{\Huge{\frown}}{PM}\)), then the shaded region is a sector. We can call it sector \(PAM\) (though the order of the letters does not matter).
b) Notice that \(\overline{BT}\) and \(\overline{BO}\) are chords, but not radii, the shaded region is not a sector.
c) Since the shaded region is bounded by two radii (\(\overline{AO}\) and \(\overline{AT}\)) and a circular arc (\(\overset{\Huge{\frown}}{TBO}\)), the shaded region is a sector. We can call it sector \(TAO\).
Area of Sector
Suppose you don’t want to find the area of the entire circle, just a part of it. We will use the following to find the area of a sector:
a) Since the shaded region is bounded by two radii (\(\overline{AP}\) and \(\overline{AM}\)) and a circular arc (\(\overset{\Huge{\frown}}{PM}\)), then the shaded region is a sector. We can call it sector \(PAM\) (though the order of the letters does not matter).
b) Notice that \(\overline{BT}\) and \(\overline{BO}\) are chords, but not radii, the shaded region is not a sector.
c) Since the shaded region is bounded by two radii (\(\overline{AO}\) and \(\overline{AT}\)) and a circular arc (\(\overset{\Huge{\frown}}{TBO}\)), the shaded region is a sector. We can call it sector \(TAO\).
Area of Sector
Suppose you don’t want to find the area of the entire circle, just a part of it. We will use the following to find the area of a sector:
Sector Area Given Circle Area
Example 2:
If the area of a circle is \(36\pi\) square cm, what is the area of a sector that is one-tenth of the area? Keep your answer in terms of \(pi\) and round your answer to the nearest tenth.
Solution:
Recall that sector area is a fraction of the circle’s total area. Since we know that we are looking for \(\Large\frac{1}{10}\) of the area, let’s calculate the following:
sector area \(= \Large\frac{1}{10}\normalsize\cdot36\pi\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=3.6\pi\) cm\(^2\;\;\) This is the answer in terms of pi
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=11.30973355...\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\approx 11.4\)cm\(^2\;\;\) This is the answer rounded to the nearest tenth.
Sector Area Given Central Angle and Radius
Example 2:
If the area of a circle is \(36\pi\) square cm, what is the area of a sector that is one-tenth of the area? Keep your answer in terms of \(pi\) and round your answer to the nearest tenth.
Solution:
Recall that sector area is a fraction of the circle’s total area. Since we know that we are looking for \(\Large\frac{1}{10}\) of the area, let’s calculate the following:
sector area \(= \Large\frac{1}{10}\normalsize\cdot36\pi\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=3.6\pi\) cm\(^2\;\;\) This is the answer in terms of pi
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=11.30973355...\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\approx 11.4\)cm\(^2\;\;\) This is the answer rounded to the nearest tenth.
Sector Area Given Central Angle and Radius
Example 3:
Given \(\bigodot N\), find the area of sector \(ONE\). Keep your answer in terms of \(\pi\). Solution: Let’s use the formula for sector area to find it: sector area \(= \Large\frac{arc\; measure}{360^{\circ}}\normalsize\cdot r^2\pi\) \(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\Large\frac{80^{\circ}}{360^{\circ}}\normalsize\cdot(3)^2\pi\) \(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\Large\frac{2}{9}\normalsize\cdot9\pi\) \(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=2\pi\) units\(^2\) |
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:
sector area \(= \Large\frac{m\overset{\Huge{\frown}}{KW}}{360^{\circ}}\normalsize\cdot r^2\pi\)
\(\;\;\;\;\;\;\;\;\;\;\;\;20=\Large\frac{60^{\circ}}{360^{\circ}}\normalsize\cdot r^2\pi\;\;\) Substitute the values that we know
\(\;\;\;\;\;\;\;\;\;\;\;\;20=\Large\frac{1}{6}\normalsize\cdot r^2\pi\;\;\) Simplify
\(\;\;\;\;(6)(20)=(6)\left(\Large\frac{1}{6}\normalsize\cdot r^2\pi\right)\;\;\) Multiply by the reciprocal
\(\;\;\;\;\;\;\;\;\;\;120=r^2\pi\)
\(\;\;\;\;\;\;\;\Large\frac{120}{\pi}=\frac{r^2\pi}{\pi}\)
\(\;\;\;\;\;\;\;\Large\frac{120}{\pi}\normalsize=r^2\)
\(\pm\sqrt{\Large\frac{120}{\pi}}\normalsize = r\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;r=\pm6.18038723...\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;r\approx 6.2\) inches length must be positive
We can also use sector area combined with what we already know about areas of other figures. We simply must break up the area into areas of shapes that we can find. Try the example below.
Example 5:
In the diagram below \(\bigodot A\cong\bigodot C\cong\bigodot L\cong\bigodot J\) and the circles are bounded by tangents. Find the area of the unshaded region.
We will still use the arc length equation, but we will need to use order of operations to find the length of the radius:
sector area \(= \Large\frac{m\overset{\Huge{\frown}}{KW}}{360^{\circ}}\normalsize\cdot r^2\pi\)
\(\;\;\;\;\;\;\;\;\;\;\;\;20=\Large\frac{60^{\circ}}{360^{\circ}}\normalsize\cdot r^2\pi\;\;\) Substitute the values that we know
\(\;\;\;\;\;\;\;\;\;\;\;\;20=\Large\frac{1}{6}\normalsize\cdot r^2\pi\;\;\) Simplify
\(\;\;\;\;(6)(20)=(6)\left(\Large\frac{1}{6}\normalsize\cdot r^2\pi\right)\;\;\) Multiply by the reciprocal
\(\;\;\;\;\;\;\;\;\;\;120=r^2\pi\)
\(\;\;\;\;\;\;\;\Large\frac{120}{\pi}=\frac{r^2\pi}{\pi}\)
\(\;\;\;\;\;\;\;\Large\frac{120}{\pi}\normalsize=r^2\)
\(\pm\sqrt{\Large\frac{120}{\pi}}\normalsize = r\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;r=\pm6.18038723...\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;r\approx 6.2\) inches length must be positive
We can also use sector area combined with what we already know about areas of other figures. We simply must break up the area into areas of shapes that we can find. Try the example below.
Example 5:
In the diagram below \(\bigodot A\cong\bigodot C\cong\bigodot L\cong\bigodot J\) and the circles are bounded by tangents. Find the area of the unshaded region.
Solution: Since the unshaded region isn’t made up of triangles, quadrilaterals, sectors, or anything that we know exactly how to find the area, we will need to take a different approach. We will find the area of the whole figure and subtract out the area of the shaded region! Let’s watch this video for its solution!
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