regular pentagon with sides 200 m in length, they build a fence 3m from the building, what is the perimeter?

Answer 1

The internal angle of a regular pentagon is 108 degrees. If you draw the fence as a regular pentagon outside of the original pentagon, and then connect the vertices of the inner pentagon to that of the outer, you'll see that you have a set of five isoceles trapezoids with base angles of 54 degrees, being half of the 108 degree internal angle of the larger polygon. Drop an altitude from the upper vertex of the trapezoid to the longer base, and you create a right triangle with one angle equal to 54 degrees, and the known length of a leg opposite that angle, 3 m because it was given as the distance from the original pentagon to the fence. Now you can easily use trigonometry to find the length of the other leg of the right triangle, which is the distance by which the fence overhangs the 200 m side on that end. There are 10 such overhangs (2 on each end of the 5 sides), so calculate the total overhang and add it to the perimeter of the original pentagon, which is simply 5*200 = 1000 m.

Note that if this had been a square, each side would have become 6 m longer, since the internal angle would have been 90 degrees, half of it would have been 45 degrees, and the overhanging triangles would be isoceles. The change here will be smaller due to the greater base angle.

Incidentally, I made certain assumptions here. The primary one was that when we say the fence is 3m from the building, we mean that the fence is a pentagon with sides parallel to the original sides of the pentagon, and 3m from them. The vertexes of the fence will actually be more than 3m from the vertexes of the original pentagon. If we didn't make this assumption, we'd have to envision the fence as being a pentagon with rounded corners, being the pentagon fence described above but with the vertices filleted with radii of 3m, a more complicated figure.

Answer 2

I guess what you're saying:
- You have a building in the shape of a regular pentagon;
- A pentagonal fence is built around it, each side of which is a farther 3 meters out than the building.

So what is the perimeter of the fence?

OK, I guess there is no way around the trigonometry:
- Each side subtends an angle (in radians) of 2pi/5
- The triangle associated with that side is an isosceles triangle; when split down the middle, there are two congruent parts, each of which is a right triangle whose height is the perpendicular to the side, and whose base is half the side; and whose vertex subtends half the angle. So, the entirety consists of 2*5 = 10 right triangles, with height h = perpendicular to the side and base = h*tan((2*pi/5)/2)
= h*tan(pi/5). But two of these bases form a side, so
200 = 2*h*tan(pi/5), or
h = 200/(2*tan(pi/5))
= 100/(0.72654)
= 137.64

Now, if you add another 3 meters to make the fence, the height of the triangular subsections is increased from:
137.64 to 140.64, a factor of (140.64/137.64) = 1.0218

Each side will be scaled up from:
200 to 200*1.0218 = 204.36
So the total perimeter of the fence will be:
5 * 20.36 = 1021.8 meters.

Answer 3

A pentagon has 5 factors. the fringe in 32 cm permit the dimensions of the two factors of the common pentagon be 'x' 5*x = 32cm x = 32/5 (Divide the two factors via 5) x = 6.40cm or 640mm The length of the two factors of the common pentagon is 6.40 cm (or) 640mm.

Answer 4

Each side becomes 6m longer, so

206 x 5 = 1030m

Answer 5

1021.8
for the little overhang
tan 36=x/3
x=2.18
1000+10(2.18)

Answer 6

6(200xroot3 + 2x3)/root3= 1200+12xroot3 =1220.785 m app..