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question is:

regular n-sided polygon has all of its vertices on a circle of radius 0.5m Find a formula, P = ... in terms of n, for the perimeter of this polygon

really need someone to solve it for me.

Answer 1

P = 2 n r sin (pi/n)
where P = Perimeter, n = number of sides of polygon, r = radius.
The proof is quite simple but difficult to explain without making figures but still I try to explain. See if a polygon has n sides it will have interior angle as 2pi/n. Now take any triangle it will have 2 sides as r and and internal angle as 2pi/n. We need to get the side of the polygon. Side of the polygon is 2 r sin(pi/n) and P = n x side of polygon i.e.
P = 2 n r sin(pi/n). Also area of a polygon is r^2 n sin(2pi/n)/2. Let me know if you don't understand anything.

Hope it helps. Read this for perimeter http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html and for area read this http://www.mathopenref.com/polygonregulararea.html

Answer 2

Hiya! Thanks for such a simple one for today. Ready? Here goes....

As your dead parrot (er - sorry, polygon) has 'n' sides, it is composed of 'n' isosceles triangles of equal size; each with its apex at the centre of the 0.5m circle and, of course, with its height being 0.5m. Each of these triangles can in turn be split down the middle to form two identical (reflected) right-angled triangles - each with a non-hypotenuse side (height) of 0.5m. With me so far?

Ok, here's the next step. Since the polygon has n sides, there are now 2n right-angled triangles. The angle subtended by each such triangle at the centre of the circle is thus 360deg / 2n, i.e. 180/n. As the height of each triangle is 0.5m, the width at the base is 0.5 Tan (180deg/n) m. The length of each side of the polygon is thus twice this amount - i.e. tan (180deg/n) m.

Now you know the length of each side of the polygon, you need only to multiply by the number of sides. Therefore the final length of the perimeter of the polygon you have described is n(Tan(180deg/n)) metres.

I hope this helps.

By the way, I am assuming here that you have used the word 'vertices', i.e. plural of vertex & thus referring to the vertical, properly. With this model, the circle is drawn INSIDE the polygon. If it is to be drawn OUTSIDE, see the following (in much the same manner as above):

The hypotenuse of each right-angled triangle is 0.5m

The length of the base is thus 0.5 Sin(180deg/n)

Consequently, simply replace Sin for Tan in the equation.

Good luck!

Answer 3

The diameter is generally named 2r.The barrier of maximum perimeter is: 2 x 3.14 x r. With r= 0.5 you get 3.14.

Generally a side of a polynom inside a circle with this diameter is 2rsin(180 deg./n)
With a triangle you get: 2 x 0.5 x sin 60= sin 60
With a square you get: 2 x 0.5 x sin 45= sin 45
General formula when r=0.5:
Perimeter= n x sin (180/n)

Answer 4

Okay, you have to look at the polygon as 2n seperate triangles. The side of the triangle that's not the radius or hypothenuse is (.5)(Cos(90-(360/2n))).
So, the formula is (n*2*.5)(Cos(90-(360/2n))) = n*(Cos(90-(360/2n))).

Answer 5

3. 80x 4. 80x + a million 5. 250(80x + a million) 6. 250(80x + a million) + y 7. 250(80x + a million) + 2y 8. 250(80x + a million) + 2y - 250 9. [250(80x + a million) + 2y - 250] / 2 = [20000x + 250 + 2y - 250] / 2 = [20000x + 2y] / 2 = 10000x + y

Answer 6

As n -> infinity, P -> 2*pi*r

Answer 7

p= 180(n-2)

Answer 8

the anser is pie

i like pie pie is always the answer

Answer 9

P = n sin (pi/n)

Answer 10

Hold on a moment...