Star Study - 10 pointed total

Draw a circle with center at O and the ten vertices lie on the circle. The angle from one vertex A to the two opposite vertices B and C is one-half the size of the central angle BOC. Five such central angles complete one 360 degrees revolution. So angle BOC = 360/5 and vertex angle BAC = 360/10. So the total of ten such vertex angles equals 10 x (360/10) = 360 degrees

First, notice that in the middle of the many lines there is a regular pentagon. The interior angles of a pentagon add up to 540 degrees, so each of the 5 angles is 540/5 = 108 degrees. Two of those angles form straight lines (180 degrees) with the two equal angles of the isosceles triangle formed by a vertex of the star and a side of the pentagon. So the two equal angles must be 180-108 = 72 degrees each. That leaves 180-(72+72) = 36 degrees for the angle at the vertex of the 10 pointed star. Since there are 10 points, there is 10*36 = 360 degrees total in the interior angles.

All polygon internal vertices' angles always sum up to 360.

notice it's just 2 stars stacked and we see it from above...

a single star have the inner pentagon, 5 point, each 108 degrees

now you know the single point at star 'hand' have 36 degrees

just multiply it by 10 ....and viola... 360 degrees is the answer

just break this diagram into two 5-pointed star. Now you can easily see that the pentagon formed inside each 5 pointed star, internal angles adds up to 540 degrees. Therefore each angle = 108. Thus the vertex of star constitute angle= 36. Thus the sum total becomes 36*10=360

it is true for all polygonal vertices internal angles..it is always 360

0 x (360/10) = 360 degrees

This is having 10 sided polygon inside whose sum of all anles is 1080,similar to 5 star angles this forms 10 star angles and their sum is 360 degree. K.K.GARG,India

Look at the image carefully and you will see that this star has been made by superimposing 2 5-pointed stars on one another. So, the angle at the vertex will be $36^{\circ}$ , as per previous problems in this set. Therefore, sum of internal angles is $\boxed{360^{\circ}}$

Any segment which tie two point of a circunsference is ¨seen¨ by the points of that circunsference by either a half of the angle seen from the center or its complementary (a diameter is always seen with a 90 degrees angle from both sides), a five points star uses a penthagonal distribution with 72 degrees from the center which means that the star internal angles are 36 degrees.