the demonstration of s = rθ .???

Question

s is the lengh of the arc (m) , r is the radius (m) et θ is the angle in radian how to demonstrate this formula??? thanks a lot

Answer 1

Draw a regular hexagon inside a circle.
Refer this site to draw regualr hexagon http://en.wikipedia.org/wiki/Hexagon
"A regular hexagon is constructible with compass and straightedge. The following is a step-by-step animated method of this, given by Euclid's Elements, Book IV, Proposition 15."
The internal angles of a regular hexagon (one where all sides and all angles are equal) are all 120° and the hexagon has 720 degrees. It has 6 lines of symmetry
Join the vertices
There are 6 equilateral triangles. The anngle subtended by each side on the center of the circle is 60 degree.
Each arc of the circle, therefore subtends 60 degree at the center. The length of each arc S is approximately equal [ actually greater ] to the radius.
One radian is by defenition the angle subtended by the arc whose length is equal to the radius.
In the figure each side of the hexagon or each arc is [approximately] equal to the radius.
Hence we can take one radian as equal to approximately to 60 degree. There are six such arcs and hence the perimeter of the circle is 6 arc lengh . or six times the radius.
Hence S = r θ where θ is the angle in radian.
But since the arc is slightly greater than the radius, and the correction to the formuls is
1 radian is not equal to 60 degree, but it is equal to 57.2957795130823208767981548141056.283185307179586476925286766559 to a great accuracy.

Now Draw a new circle and draw a horizontal radius on the positive x direction. This radius is the first one.
Do not draw the diameter.. Draw another radius[ no 2 ] which is exactly at 57 degree to the previous diameter. This should be done with great accuracy. A big circle will enable us to do this. Similarly draw six radii each at angle of 57 degree.
The total angle covered is 6x 57 = 342 degree. The remaining angle is 18 degree.
Since each arc equals the radius to a great accuracy, now we have 6 arc length and another are whose length corresponds to 18 degree.
Thus the circumference of the circle is more than 6 radian.
The more accurate value is 6.283185307179586476925286766559 radian.

Answer 2

Before jumping into the question, some background knowledge is required.
How do you define radian? One radian = angle subtended at the center of the circle when the arc length is numerically equals to the radius of the circle. Thus, the radian subtened at the center of the circle is defined as the ratio of the arc length to its radius.
Hence, θ=s/r
Re-arranging, s=rθ (shown)

Answer 3

i'm sure you're familiar with the perimeter of a circle equation: p = 2*pi*r
where p is the perimeter of a circle, 2*pi is the angle of one full rotation around the circle, r is the radius.

it's essentially derived from s=rθ, where θ =2*pi.

Answer 4

Suppose you have a circle of radius r. How many radians is there in one complete circle? What is s for a complete circle? You now have your answer.