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Perimeter of a Sector

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Geometry is one the important branches of mathematics concerned with various geometric shapes or objects. Also deal with the problems such as size, relative position of shapes, area, volume, lengths and other properties of space. Also include formulas for areas, lengths and volumes, such as area of a circle, area of sector, area of a triangle, circumference, perimeter, the Pythagorean theorem and volume of a cone, cuboid, sphere, cylinder and of course volume of pyramid. A sector is a kind of two-dimensional geometric shape that arises from a small piece of a circle. Sector of a circle is a part of a circle out of 360 degrees. In this section will study about how to find the perimeter of a sector of a circle with couple of solved examples.

Perimeter of a Sector Circle

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Circle is the set of points in a plane that are at a constant distance from the center. A perimeter is a total path around a two dimensional figure. Perimeter of any polygon can be calculated by adding all the sides measures. In case of ellipse or circle it is called as circumference. 

To find the perimeter of sector of circle follow the below steps:

Suppose a circle with radius "a". 
Step 1: Divide the circle by its diameter. Shaded part of the circle, bounded by the circumference and two radii is a sector of a circle or we can say semicircle is a sector. Shown below:

Perimeter of a Sector Circle
Step 2: Find the perimeter of above figure (shown in step 1):

Length of the curved edge of the sector = $\frac{1}{2}$(Perimeter of whole circle)
Perimeter of circle = $2\pi\ a$
So length of the curved edge = $\frac{1}{2}$$(2 \pi a)$ = $\pi a$

Perimeter of Sector
Step 3: Find perimeter of semicircle:

We know that, Perimeter of any object is the sum of its all edges.

= Perimeter of semicircle = Length of curved edge + Length of straight edge

= $\pi\ a\ +\ a\ +\ a$

= $\pi\ a\ +\ 2a$
Step 4: Find perimeter of sector 

Consider a quarter circle (Half of the semicircle), shown below

Sector Perimeter

Length of selected curve = $\frac{\pi a}{2}$

Perimeter = $\frac{\pi a}{2}$ + a + a = $\frac{\pi a}{2}$ + 2a
Step 5: Perimeter of any sector with angle x.

As same as above, we can calculate the perimeter of any sector

Measure Perimeter of a Sector
A sector of $x^0$ = $\frac{x}{360}$

Length of the curved edge =
$\frac{x}{360}$ $\times$ 2 $\pi$ a = $\frac{\pi \times a}{180}$

Therefore, the perimeter of any sector = 2a +
$\frac{\pi \times a}{180}$

Perimeter of a Sector Formula

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 A sector is bounded by means of a couple immediately outlines add up to this radius with the group as well as a round arc. The angle involving the two radii would be the angle of your sector. 
Consider a circle have radius r and angle of the sector is x degrees, then

Perimeter of any sector = $2r$ + $\frac{\pi \times r}{180}$.

Where $\pi$ is a constant add up to 3. 14159... as well as x (sector angle) is in degrees. 
If angle,x, is measured in radians, then the length of the arc found by using the equation, L = ar.

How to Calculate Perimeter of a Sector

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Follow the below steps to calculate the perimeter of a sector:

Step 1: Calculate the measure of radius and angle of sector (if not given directly).

Step 2:
Substitute all the values in the formula (mentioned in above section).

Step 3:
Reduce the expression, if possible. Write your answer in units.
Below is an example to understand more better:

Example: 

Find the perimeter of the below figure having radius $15$:

Perimeter of a Sector Example

Solution: 

Since OB(radius, r) = $15$ and x$^0$ (angle of sector) = $54^0$

Using perimeter of sector formula, we obtain

Perimeter = $2r$ + $\frac{\pi \times r}{180}$ 

Perimeter = $2 \times 15$ + $\frac{\pi \times 54 \times 15}{180}$

Perimeter = $30$ + $\frac{9 \pi }{2}$.

Perimeter of a Sector Examples

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Few examples on perimeter of sector are as follow:
Example 1: 

Find the perimeter of a sector if angle of sector of the circle is 50 degrees and the radius is 10 cm.

Solution:

Let $\theta$ and r be the angle of sector and radius of circle respectively.

Perimeter = $2r$ + $\frac{\pi \theta r}{180}$

Perimeter = $2 \times 10$ + $\frac{\pi \times 50 \times 10}{180}$

Perimeter = $20 + 2.78\pi$

Therefore perimeter of sector is $20 + 2.78\pi$ cm.
Example 2: 

Radius and the area of a sector of a circle are $6$cm and $30$sqcm. Calculate the perimeter of the sector.

Solution: 

Radius $(r)\ =\ 6$cm

and area of a sector = $30$sq cm

We know that, the formula for the area of a sector is:

A = $\frac{n}{360}$ $(\pi r^2)$ (where n = central angle and r = radius of circle)

30 = $\frac{n}{360}$ $(\pi 6^2)$

or n = $\frac{300}{\pi}$.

Now, Perimeter = $12$ + $\frac{\pi \times 300 \times 6}{\pi \times 180}$

Perimeter = $12 + 10 = 22$

Perimeter of sector = $22$cm.