Question 979530
{{{drawing(400,700/3,-1.2,1.2,-.2,1.2,
arc(0,0,2,-2,20,160),line(0,0,cos(20*pi/180),sin(20*pi/180)),
line(0,0,-cos(20*pi/180),sin(20*pi/180)),
red(arc(0,0,.3,-.3,20,160)),locate(-.1,.27,"140°"),
locate(.4,.25,r)

 )}}}
<pre>
The whole figure is a sector of a circle. The area of a sector of a circle
is given by the formula:

{{{A}}}{{{""=""}}}{{{expr(1/2)theta*r^2}}} where <font face="symbol">q</font> is in radians.

We convert <font face="symbol">q</font> = 140° to radians:

{{{"140°"*pi/"180°"}}}{{{""=""}}}{{{7pi/9}}}

{{{126pi}}}{{{""=""}}}{{{(expr(1/2)*7pi/9)*r^2}}}

{{{126pi}}}{{{""=""}}}{{{(7pi/18)*r^2}}}

Multiply both sides by 18

{{{2268pi}}}{{{""=""}}}{{{7pi*r^2}}}

Divide both sides by 7<font face="symbol">p</font>

{{{324}}}{{{""=""}}}{{{r^2}}}

{{{18}}}{{{""=""}}}{{{r}}}

{{{18}}}{{{""=""}}}{{{r}}}

The length of the arc is found by the formula

{{{s}}}{{{""=""}}}{{{r*theta}}} where <font face="symbol">q</font> is in radians.

{{{s}}}{{{""=""}}}{{{18*(7pi/9)}}}

{{{s}}}{{{""=""}}}{{{14pi}}}

The perimeter is made up of two radii and the arc at the top:

Perimeter = {{{36+14pi}}}

Edwin</pre>