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Hello! Let's say we have an arc, and its measure is 36 degrees. If we know that one side length is 10 units what is the perimeter of the arc?

ant101 Dec 2, 2018

#2**+2 **

Ant....I assume that we have a decagon [ regular polygon of 10 sides ] inscribed in a circle??......if so....

The radius, r, of the circle can be found as

r / sin 72 = 10/ sin 36

r = 10sin72 / sin 36 = 5 (1 + √5 )

So....the perimeter of the arc [ arc length of 1/10 of the circle ] is

2pi r / 10 =

2pi [ 5 ( 1 + √5)] / 10 = pi * (1 + √5) units ≈ 10.17 units

CPhill Dec 2, 2018

#5**+2 **

Oh, OK....the * radius* = 10

Arc length = arc length of whole circle * (36/360) =

2pi (radius) * (36/360) =

2pi (10) * (1/10 ) =

2pi units ≈ 6.28 units

CPhill
Dec 2, 2018

#7**+2 **

Thanks CPhill, since you are not here, and ant101 has asked for clarifictaion, I will add my input.

Hi Ant101

There are 360 degrees in a circle

so 36 degrees is one tenth of a circle and the radius of your circle is 10

so the arc length is \(\text{one tenth }*\pi*10^2 = 0.1*\pi*100=10\pi\)

NOW, your question is worded incorrectly and technically it does not make sense.

An arc is a part of a circle's circumference and it does not have a perimeter because it is not a closed curve.

An arc has a length, which I have alredy found for you.

I think you are actually asking for the perimeter of the sector.

A sector is a part of a circle which includes 2 radii and the arc between them.

For instance, when you cut a cake you usually cut it into sectors.

so

The perimeter of the sector is \(10\pi+10+10 = 20+10\pi\;\;units\)

-----------------------

My answer is very different from CPhill's. This is because the question is worded incorrectly and he interpreted its meaning very differently from me.

Melody
Dec 2, 2018

#8**+2 **

Ah...I see....Ant wanted the perimeter of the* sector*.....!!!

This should be

Sum of radii + arc we found

10 + 10 + 2pi = [20 + 2pi ] = 26.28 units

And you are correct, Ant....

CPhill
Dec 2, 2018

#9**+3 **

I think you made a slight error, Melody

You computed (1/10) of the * area *of the circle...not (1/10) of the circumference

CPhill
Dec 2, 2018