# What is the perimeter of a sector?

I don't understand this. So we have:

\begin{align} r &= 12 \color{gray}{\text{ (radius of circle)}} \\ d &= 24 \text{ (r}\times2) \color{gray}{\text{ (diameter of circle)}} \\ c &= 24\pi \text{ (}\pi\times d) \color{gray}{\text{ (circumference of circle)}} \\ a &= 144\pi \text{ (}\pi\times r^2) \color{gray}{\text{ (area of circle)}} \end{align}

And we have:

\begin{align} ca &= 60^\circ \color{gray}{\text{ (Central Angle of sector)}} \\ ratio &= \frac{60}{360} = \frac{1}{6} \color{gray}{\text{ (ratio of ca to circle angle which is 360 degrees)}} \end{align}

So now we can calculate:

\begin{align} al = \frac{1}{6} \times 24\pi &= 4\pi \color{gray}{\text{ (arc length of SECTOR = ratio X circumference of circle)}} sa &= \frac{1}{6} \times 144per = 24\pi \color{gray}{\text{ (sector area = ratio X area of circle)}} \end{align}

So my question is: What is meant by the perimeter of a Sector. Is it the arch length or the are of a Sector? And what is $$24 + 4\pi$$?

• "Perimeter" always means length, not area. – Andreas Blass May 18 '13 at 20:01
• @AndreasBlass: so perimeter of a sector = arc length of a sector? Than why + r X 2? – Jawad May 18 '13 at 20:04
• @Jawad. Because the sector has also two sides equal to radius. – Jean-Claude Arbaut May 18 '13 at 20:06
• @AndreasBlass : Thanks mate – Jawad May 18 '13 at 20:18
• @arbautjc : Thanks man – Jawad May 18 '13 at 20:19

## 3 Answers

The perimeter of the sector includes the length of the radius $\times 2$, as well as the arc length. So the perimeter is the length "around" the entire sector, the length "around" a slice of pizza, which includes it's edges and its curved arc.

The arc length is just the curved portion of the circumference, the sector permimeter is the length of line $\overline{AC} = r$ plus the length of line $\overline{BC} = r$, plus the length of the arc ${AOC}$.

The circumference of the circle is the total arc length of the circle.

Length is one-dimensional, the length of a line wrapped around the circle. Area is two dimensional; All of what's inside the circle.

• sorry but the perimeter of a circle = circumference of the circle and should not the permimeter of a sector = arc length of the sector? – Jawad May 18 '13 at 20:02
• @Jawad. Just a question of convention (in the same vein: for area of a cylinder, do you count the two disks?), but it appears the usual convention is counting the radius (see here for example). Anyway it would be useless to have another name for arc length. – Jean-Claude Arbaut May 18 '13 at 20:05
• No, they are including the distance around the entire sector, the length "around" a slice of pizza, which includes more than just the outer arc length. The arc length is just the curved portion of the circumference, the sector permimeter is the length line $AC$ plus the length of line $BC$ plus the length of the arc $AOC$. – amWhy May 18 '13 at 20:06
• so the perimeter of a CIRCLE = circumference of that circle BUT the perimeter of a SECTOR = arc length of that circle + r X 2? Right? – Jawad May 18 '13 at 20:09
• Yes, exactly. That's correct. – amWhy May 18 '13 at 20:11

perimeter of the sector is the sum of the lengths of all its boundaries.thus the perimeter of the sector is L+2r units.

• Your answer could be clarified by specifying what those boundaries are. In particular, you should indicate how your formula leads to the correct answer. – N. F. Taussig Mar 11 '15 at 13:57

$\frac{\theta}{360}\times 2\pi (r) + 2r$ That would be the length of the arc in the first part and twice the radius in the second part.

• Welcome to this site! The question you just answered is rather old and already has perfectly good answers, to which your post doesn't seem to add much. More value to the site would be added by answering unanswered questions, or providing answers with significantly new information. – Pierre-Guy Plamondon Jun 29 '15 at 9:58