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Circle

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Sector Calculator

rotating circle

sector approximations

Area:
Perimiter:
arc:

chord:

radius:

angle:

Enter two values (atleast one being an angle or radius)

area:
a:
c:
A: °
r:
sector image

Sector Tutorial

The formulas for sectors are shown below (assuming angle value is in degrees):

\text{arc}=\text{radius}\cdot \text{angle}\cdot \frac{\pi }{180}

\text{chord}=2\cdot \text{radius}\cdot \sin (\frac{\text{angle}\cdot \frac{\pi }{180}}{2})

\text{area}=\frac{\text{radius}^2\cdot \text{angle}\cdot \frac{\pi }{180}}{2}

\text{perimiter}=2\cdot \text{radius}+\text{arc}

\text{radius}=\frac{\text{chord}}{2\cdot \sin \frac{\text{angle}\cdot \frac{\pi }{180}}{2}}

\text{angle}=\frac{\text{arc}}{\text{radius}}\cdot \frac{180}{\pi }



Sectors are portoins of a circle with the four parts being the central angle, radius, arc length, and chord length. The central angle of a sector is the angle that substends at the center of the circle to two points on a cicle. The perimiter between the two points on the circle is called the arc of the sector. The chord length is the shortest distance between the two points on the circle and the radius is the distance from the center of the circle to any point on the circle's perimiter. Given either one angle value and any other value or one radius length and any other value, all unknown values of a sector can be calculated. Without either a radius length or angle measure, dimensions of a sector are not calculatable. If told to find the missing values of a sector given a radius of length 34 and an arc of length 38, all other unknown values can be calculated. The first step in calculating these values is finding out which value must be found first. If given a radius length, the angle must be calculated first and if given an angle, the radius must be calculated first. The calculation process of finding the angle is shown below:

\text{angle}=\frac{\text{arc}}{\text{radius}}\cdot \frac{180}{\pi }

\text{angle}=\frac{(38)}{(34)}\cdot \frac{180}{\pi }

\text{angle}=\frac{3420}{(17\pi )}^{\circ }

\text{(in simplest form)}


Now that the angle is known, we can find all other values of the sector:

calculating the chord length:

\text{chord}=2\cdot \text{radius}\cdot \sin (\frac{\text{angle}\cdot \frac{\pi }{180}}{2})

\text{chord}=2\cdot (34)\cdot \sin (\frac{(\frac{3420}{(17\pi )})\cdot \frac{\pi }{180}}{2})

\text{chord}=68\sin (\frac{19}{34})

\text{(in simplest form)}

calculating the area:

\text{area}=\frac{\text{radius}^2\cdot \text{angle}\cdot \frac{\pi }{180}}{2}

\text{area}=\frac{(34)^2\cdot (\frac{3420}{(17\pi )})\cdot \frac{\pi }{180}}{2}

\text{area}=646

\text{(in simplest form)}

calculating the perimiter:

\text{perimiter}=2\cdot \text{radius}+\text{arc}

\text{perimiter}=2\cdot (34)+(38)

\text{perimiter}=106

\text{(in simplest form)}

Tutorial given angle



If told to find the missing values of a sector with angle 58 degrees and chord of length 43, you can begin solving by finding the radius length. As stated above, if given an angle value with an unkown radius length, it is the radius that must be calculated before anything else. To find the radius length given the chord length and angle measure you can follow the calculation process shown below:

\text{radius}=\frac{\text{chord}}{2\cdot \sin \frac{\text{angle}\cdot \frac{\pi }{180}}{2}}

\text{radius}=\frac{(43)}{2\cdot \sin \frac{(58)\cdot \frac{\pi }{180}}{2}}

\text{radius}=\frac{43}{(2\sin (\frac{29\pi }{180}))}

\text{(in simplest form)}



calculating the arc length:

\text{arc}=\text{radius}\cdot \text{angle}\cdot \frac{\pi }{180}

\text{arc}=(\frac{43}{(2\sin (\frac{29\pi }{180}))})\cdot (58)\cdot \frac{\pi }{180}

\text{arc}=\frac{1247\pi }{(180\sin (\frac{29\pi }{180}))}

\text{(in simplest form)}



calculating the area:

\text{area}=\frac{\text{radius}^2\cdot \text{angle}\cdot \frac{\pi }{180}}{2}

\text{area}=\frac{(\frac{43}{(2\sin (\frac{29\pi }{180}))})^2\cdot (58)\cdot \frac{\pi }{180}}{2}

\text{area}=\frac{53621\pi }{(720\sin (\frac{29\pi }{180})^2)}

\text{(in simplest form)}



calculating the perimiter:

\text{perimiter}=2\cdot \text{radius}+\text{arc}

\text{perimiter}=2\cdot (\frac{43}{(2\sin (\frac{29\pi }{180}))})+(\frac{1247\pi }{(180\sin (\frac{29\pi }{180}))})

\text{perimiter}=\frac{(\frac{1247\pi }{180}+43)}{\sin (\frac{29\pi }{180})}

\text{(in simplest form)}



Sectors can be calculated given a variety of different dimension but the above examples show just two of them.