<p>This is from the january 2006 sat,
In the figure above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If x + y = 80, how many sides does the polygon have?
Yfrog</a> - lmy.jpg</p>
<p>alos there is this
A sphere with a r radius r inside a tube touches each one of the six sides of the cubs. What is the volume of the cube in terms of r?</p>
<p>Thanks guys</p>
<p>The shown figure is a quadrilateral (4 sides), so the sum of all its angles’ measures is 360 degrees.</p>
<p>x+y = 80
So the sum of the other 2 angles’ measures is 360 - 80, or 280.
Since the other 2 angles have equal measures (as stated in the problem), the measures must be 140.</p>
<p>DOING IT BY HAND:
Regular 3-sided shapes have 60 degree angles - 180 in total
Regular 4-sided shapes have 90 degree angles - 360 in total
Regular 5-sided shapes have ___ degree angles - 540 in total
Regular 6-sided shapes have ___ degree angles - 720 in total
Regular 7-sided shapes have ___ degree angles - 900 in total
Regular 8-sided shapes have ___ degree angles - 1080 in total
Regular 9-sided shapes have ___ degree angles - 1260 in total</p>
<p>140 is the total divided by the number of sides (1260/9), so the answer is 9 sides</p>
<p>You keep adding 180 as the total number of sides increases by 1</p>
<p>DOING IT BY EQUATION:
sum of all angles’ measures =180(n-2), where n = number of sides
180(3-2)=180
180(4-2)=360
180(5-2)=540
180(6-2)=720
180(7-2)=900
180(8-2)=1080
180(9-2)=1260</p>
<p>9 angles of 140 degrees each = 1260</p>
<p>Or, continuing from the point when you have found the interior angles to be 140…</p>
<p>If the interior angle is 140, then the exterior angle is 40 (bc 40+140=180).
Since the exterior angles add up to 360, no matter how many sides, there must be 360/40=9 of them.</p>
<p>Thanks guys this makes a lot of sense now</p>
<p>As for the radius and cube question…</p>
<p>Basically the sphere sits inside the cube. The diameter is actually what dictates the length of one side of the cube. Thus, Volume of the cube= l x w x h, but since all are equal within a cube it becomes l^3 or Diameter of the sphere^3 (since diameter=length of one side). Diameter of the sphere = 2r</p>
<p>so we have…</p>
<p>Volume= 2r^3= 8r^3.</p>
