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Circle Arc, segment, sector Torus Example 1-Sigment

Circle calculator

circle
Circle
* Input a value in any field
   
Radius (R)
Diameter (D)
Circumference (P)
Area (A)
Circle equations summary

Arc, Segment, Sector Calculator and equations

Arc
Arc, Segment, Sector Calculator ▲
Arc, Segment and Sector
* Input 2 values in any unmarked fields
   
Radius (r)
Central angle (θ)
Arc length (L)
Sector area
Sector circumference
Segment area
Segment circumference
Triangle base (c)
Segment height (h)
Triangle height (t)
Input limit:
Input limit:
Arc, segment and sector equations summary
Arc, segment and sector equations summary

Torus calculator and equations

Torus
Torus ▲
Torus
* Input 2 values in any unmarked field
   
Outer radius (R)
Inner radius (r)
Ring radius (a)
Mean radius (c)
Torus volume (V)
Torus surface Area (S)
Input limit:
Torus volume
S = 4π2a c = π2(R + r)(R − r) = π2(R2 − r2)
R = c + a                             r = c − a
Torus geometry
Example 1 - segment area Print line equations ▲
Find the expression for the area of a segment defined by the radius  r  and the segment height  h.
Segment example
The area of a sector whose angle equals to θ is:
A_sector=θ/2 r^2 (θ in radians)
The area of the triangle formed by the two radii and
the cord  c  is: A_triangle=t c/2
The value of  t  is: t = r - h
The value of  c  using the Pythagoras theorem is:
c=2√(r^2-t^2 )=2√(r^2-(r-h)^2 )=2√(2rh-h^2 )
θ is: θ=2 cos^(-1)⁡(t/r)=2 cos^(-1)⁡((r-h)/r)
Now the area of the segment is:
A_segment=A_sector-A_triangle=θ/2 r^2-t c/2=θ/2 r^2-(r-h) √(2rh-h^2 )
And after substituting the value of  θ  we get the final value:
A_segment=r^2 〖cos〗^(-1)⁡((r-h)/r)-(r-h) √(2rh-h^2 )
Notice that when   r = h   (segment is half circle)   Asegment = r2cos-1 0 = r2 π / 2   this is half circle area.