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# Area of Sectors and Segments

## Area of parts of a circle.

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Area of Sectors and Segments

What if you wanted to find the area of a pizza, this time taking into consideration the area of the crust? Remember, crust typically takes up some area on a pizza. Leave your answers in terms of π\begin{align*}\pi\end{align*} and reduced improper fractions.

a) Find the area of the crust of a deep-dish 16 in pizza. A typical deep-dish pizza has 1 in of crust around the toppings.

b) A thin crust pizza has 12\begin{align*}\frac{1}{2}\end{align*} - in of crust around the edge of the pizza. Find the area of a thin crust 16 in pizza.

c) Which piece of pizza has more crust? A twelfth of the deep dish pizza or a fourth of the thin crust pizza?

### Area of Sectors and Segments

A sector of a circle is the area bounded by two radii and the arc between the endpoints of the radii.

The area of a sector is a fractional part of the area of the circle, just like arc length is a fractional portion of the circumference. The Area of a sector is A=mABˆ360πr2\begin{align*}A= \frac{m\widehat{AB}}{360^\circ} \cdot \pi r^2\end{align*} where r\begin{align*}r\end{align*} is the radius and ABˆ\begin{align*}\widehat{AB}\end{align*} is the arc bounding the sector. Another way to write the sector formula is A=central angle360πr2\begin{align*}A=\frac{central \ angle}{360^\circ} \cdot \pi r^2\end{align*}.

The last part of a circle that we can find the area of is called a segment, not to be confused with a line segment. A segment of a circle is the area of a circle that is bounded by a chord and the arc with the same endpoints as the chord. The area of a segment is Asegment=AsectorAABC\begin{align*}A_{segment}=A_{sector}-A_{\triangle ABC}\end{align*}

#### Finding the Area in Terms of Pi

Find the area of the blue sector. Leave your answer in terms of π\begin{align*}\pi\end{align*}.

In the picture, the central angle that corresponds with the sector is 60\begin{align*}60^\circ\end{align*}. 60\begin{align*}60^\circ\end{align*} would be 16\begin{align*}\frac{1}{6}\end{align*} of 360\begin{align*}360^\circ\end{align*}, so this sector is 16\begin{align*}\frac{1}{6}\end{align*} of the total area.

area of blue sector=16π82=323π\begin{align*}area \ of \ blue \ sector=\frac{1}{6} \cdot \pi 8^2=\frac{32}{3} \pi\end{align*}

#### Calculating the Area

Find the area of the blue segment below.

As you can see from the picture, the area of the segment is the area of the sector minus the area of the isosceles triangle made by the radii. If we split the isosceles triangle in half, we see that each half is a 30-60-90 triangle, where the radius is the hypotenuse. Therefore, the height of ABC\begin{align*}\triangle ABC\end{align*} is 12 and the base would be 2(123)=243\begin{align*}2 \left( 12 \sqrt{3} \right)=24 \sqrt{3}\end{align*}.

Asector=120360π242=192πA=12(243)(12) =1443\begin{align*}A_{sector} &= \frac{120}{360} \pi \cdot 24^2 && A_{\triangle} =\frac{1}{2} \left( 24 \sqrt{3}\right)(12)\\ &= 192 \pi && \quad \ = 144\sqrt{3}\end{align*}

The area of the segment is A=192π1443353.8\begin{align*}A=192 \pi - 144 \sqrt{3} \approx 353.8\end{align*}.

The area of a sector of circle is 50π\begin{align*}50 \pi\end{align*} and its arc length is 5π\begin{align*}5 \pi\end{align*}. Find the radius of the circle.

First substitute what you know to both the sector formula and the arc length formula. In both equations we will call the central angle, “CA\begin{align*}CA\end{align*}.”

50π5036018000=CA360πr2=CAr2=CAr2 5π=CA3602πr5180=CAr900=CAr\begin{align*}50 \pi &= \frac{CA}{360} \pi r^2 && \quad \ 5 \pi =\frac{CA}{360} 2 \pi r\\ 50 \cdot 360 &= CA \cdot r^2 && 5 \cdot 180=CA \cdot r\\ 18000 &= CA \cdot r^2 && \quad 900=CA \cdot r\end{align*}

Now, we can use substitution to solve for either the central angle or the radius. Because the problem is asking for the radius we should solve the second equation for the central angle and substitute that into the first equation for the central angle. Then, we can solve for the radius. Solving the second equation for CA\begin{align*}CA\end{align*}, we have: CA=900r\begin{align*}CA=\frac{900}{r}\end{align*}. Plug this into the first equation.

1800018000r=900rr2=900r=20\begin{align*}18000 &= \frac{900}{r} \cdot r^2\\ 18000 &= 900r\\ r &= 20\end{align*}

#### Pizza Problem Revisited

The area of the crust for a deep-dish pizza is 82π72π=15π\begin{align*}8^2 \pi - 7^2 \pi =15 \pi\end{align*}. The area of the crust of the thin crust pizza is 82π7.52π=314π\begin{align*}8^2 \pi - 7.5^2 \pi =\frac{31}{4} \pi\end{align*}. One-twelfth of the deep dish pizza has 1512π\begin{align*}\frac{15}{12} \pi\end{align*} or 54π in2\begin{align*}\frac{5}{4} \pi \ in^2\end{align*} of crust. One-fourth of the thin crust pizza has 3116π in2\begin{align*}\frac{31}{16} \pi \ in^2\end{align*}. To compare the two measurements, it might be easier to put them both into decimals. 54π3.93 in2\begin{align*}\frac{5}{4} \pi \approx 3.93 \ in^2\end{align*} and 3116π6.09 in2\begin{align*}\frac{31}{16} \pi \approx 6.09 \ in^2\end{align*}. From this, we see that one-fourth of the thin-crust pizza has more crust than one-twelfth of the deep dish pizza.

### Examples

#### Example 1

The area of a sector is 135π\begin{align*}135 \pi\end{align*} and the arc measure is 216\begin{align*}216^\circ\end{align*}. What is the radius of the circle?

Plug in what you know to the sector area formula and solve for r\begin{align*}r\end{align*}.

135π13553135225=216360πr2=35r2=r2=r2r=225=15\begin{align*}135 \pi &= \frac{216^\circ}{360^\circ} \cdot \pi r^2\\ 135 &= \frac{3}{5} \cdot r^2\\ \frac{5}{3} \cdot 135 &= r^2\\ 225 &= r^2 \rightarrow r=\sqrt{225}=15\end{align*}

#### Example 2

Find the area of the shaded region. The quadrilateral is a square.

The radius of the circle is 16, which is also half of the diagonal of the square. So, the diagonal is 32 and the sides would be 32222=162\begin{align*}\frac{32}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}=16 \sqrt{2}\end{align*} because each half of a square is a 45-45-90 triangle.

AcircleAsquare=162π=256π=(162)2=2562=512\begin{align*}A_{circle} &= 16^2 \pi =256 \pi\\ A_{square} &= \left(16 \sqrt{2} \right)^2 = 256 \cdot 2=512\end{align*}

The area of the shaded region is 256π512292.25\begin{align*}256 \pi -512 \approx 292.25\end{align*}

#### Example 3

Find the area of the blue sector of A\begin{align*}\bigodot A\end{align*}.

The right angle tells us that this sector represents 14\begin{align*}\frac{1}{4}\end{align*} of the circle. The area of the whole circle is A=π82=64π\begin{align*}A=\pi 8^2=64\pi\end{align*}. So, the area of the sector is 1464π=16π\begin{align*}\frac{1}{4}64\pi = 16\pi\end{align*}.

### Review

Find the area of the blue sector or segment in A\begin{align*}\bigodot A\end{align*}. Leave your answers in terms of π\begin{align*}\pi\end{align*}. You may use decimals or fractions in your answers, but do not round.

Find the central angle of each blue sector. Round any decimal answers to the nearest tenth.

1. The area of a sector of a circle is 54π\begin{align*}54 \pi\end{align*} and its arc length is 6π\begin{align*}6\pi\end{align*}. Find the radius of the circle.
2. Find the central angle of the sector from #13.
3. The area of a sector of a circle is 2304π\begin{align*}2304 \pi\end{align*} and its arc length is 32π\begin{align*}32 \pi\end{align*}. Find the central angle of the sector.

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### Vocabulary Language: English Spanish

chord

A line segment whose endpoints are on a circle.

diameter

A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.

pi

(or $\pi$) The ratio of the circumference of a circle to its diameter.

Arc

An arc is a section of the circumference of a circle.

arc length

In calculus, arc length is the length of a plane function curve over an interval.

A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.

Scale Factor

A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.

Sector

A sector of a circle is a portion of a circle contained between two radii of the circle. Sectors can be measured in degrees.

Sector of a Circle

A sector of a circle is the area bounded by two radii and the arc between the endpoints of the radii.