Expert: Josh - 7/24/2006

Question
Josh:
A quick question regarding finding the area of an odd figure:
If a straight line bisects a circle, you can find the area of the two half-circle shaped areas by simply finding the area of the original circle and dividing by half.
Suppose, however, you start out with that type of figure...a half circle with a straight line base.  Then another straight line (parallell to the straight-line base) bisects the half-circle, and you're left with a straight line base and a curved arc that's NOT a full half circle.  
Is there a formula for finding the area of that type of figure?
Any suggestions, assistance greatly appreciated.
Warm regards,
David Gardner

Answer
Dear David,

I understand your question perfectly. It's a simple question. But finding an expression for the area requires a fair bit of calculus. I will derive the expression using some trigonometry and borrow ideas from integration. If you just want a formula for the area, then skip the derivation and go to the last line.


.............E
.
.
..C..........F..........D
.
A............O............B

Let AB be the diameter of the circle. By construction, the length of segment AO, BO and OE are all the same. |AO|=|BO|=|OE|=r (viz., all equal to the radius).

The equation for a circle is x^2+y^2=r^2. For the right quadrant of the circle (bound by BDEO), the distance FD is generally described by x=sqrt(r^2-y^2), for 0<=x<=r.

If we consider a slice FD with infinitesimal height, dy, its area will be given by x*dy=sqrt(r^2-y^2)dy.

Now, suppose the height of OF is given by |OF|=Z. Sweeping the parameter dy from y=Z to y=r (consider the area contributed by many such slices with width x and height dy, spanning between point F and point E), we come up with the integral which describes the area bound by DEF. By left/right symmetry, this is half the area bound by DECFD, which we denote by "A".

I'll just work out the algebra now, without further explanation. You can ask me if you are interested.

A/2 = int sqrt(r^2-y^2) dy, limits taken from y=Z to y=r
   = int r sqrt(1-sin(w)^2) r cos(w) dw ...[#1],[#2]
   = r^2 int cos(w)^2 dw, limits taken from w=arcsin(Z/r) to w=pi/2
   = r^2 {int 0.5*(1+cos(2w))} ...[#3]
   = 0.5 r^2 [w+0.5sin(2w)]; from w=arcsin(Z/r) to w=pi/2
   = 0.5 r^2 [pi/2 -0 -(arcsin(Z/r)+0.5sin(2*arcsin(Z/r)))]

[#1] we let y=r sin(w), dy=r cos(w) dw
[#2] integration limits after change of variable goes from w=pi/2 to w=arcsin(Z/r)
[#3] using trig. identity cos(w)^2=[1+cos(2w)]/2

Therefore, the area of the wedge bound by DECFD is given by
A=r^2*[pi/2 - {arcsin(Z/r)+0.5*sin(2*arcsin(Z/r))}] ...[*]

The area changes, of course, depending on distance Z, i.e., how far the cut is from the center of the circle.

Note: In your problem, the radius "r" and offset distance from the center "Z" are both known. arcsin(w) is the "inverse sine" function, which appears on all scientific calculators used by high school students. It looks something like sin^(-1), where the part "-1" is written as superscript.

e.g., Let us consider some extreme cases:
case 1: If offset from center is zero, then, Z=0.
From the formula [*],
A=r^2[pi/2-{arcsin(Z/r)+0.5*sin(2*arcsin(Z/r))}]
=0.5 pi*r^2 {arcsin(0/r)+0.5*sin(2*arcsin(0))}
=(1/2) pi * r^2
As expected, this gives the area for a semi-circle

case 2: If offset from center is r, then, Z=r.
(we end up with nothing; let's see if the formula makes sense)
From the formula [*],
A=r^2[pi/2-{arcsin(Z/r)+0.5*sin(2*arcsin(Z/r))}]
=r^2[pi/2-{arcsin(1)+0.5*sin(2*arcsin(1))}]
=r^2[pi/2-{arcsin(1)+0.5*sin(2*pi/2)}] ...since sin(pi/2)=1
=r^2[pi/2-{pi/2+0.5*sin(pi)}]
=r^2[pi/2-{pi/2+0}]
=0

As expected, the equation yields zero area if we are effectively cut parallel to AB passing tangentially at point E.

Note: I can help you evaluate this value, if you give me the numbers. Don't hesitate to ask. But you can probably compute this using "=asin(w)" in Microsoft Excel spreadsheet; where "w" is replaced with some value (Z/r) between +1 and -1.

e.g., if Z=0.25r (i.e., distance of offset is equiv. to 25% of radius), the area is around 68.5% relative to a semi-circle.

Here is a table, where the area of the wedge (as you have described) is computed as a percentage of the area in a semi-circle. The value of Z varies from 0 to "r" in 1% increments (0.01*r).

Z=0.00*r: 100.000000
Z=0.01*r: 98.726782
Z=0.02*r: 97.453691
Z=0.03*r: 96.180854
Z=0.04*r: 94.908400
Z=0.05*r: 93.636456
Z=0.06*r: 92.365149
Z=0.07*r: 91.094607
Z=0.08*r: 89.824959
Z=0.09*r: 88.556333
Z=0.10*r: 87.288857
Z=0.11*r: 86.022661
Z=0.12*r: 84.757874
Z=0.13*r: 83.494627
Z=0.14*r: 82.233048
Z=0.15*r: 80.973270
Z=0.16*r: 79.715424
Z=0.17*r: 78.459642
Z=0.18*r: 77.206056
Z=0.19*r: 75.954800
Z=0.20*r: 74.706008
Z=0.21*r: 73.459815
Z=0.22*r: 72.216357
Z=0.23*r: 70.975771
Z=0.24*r: 69.738194
Z=0.25*r: 68.503764
Z=0.26*r: 67.272623
Z=0.27*r: 66.044909
Z=0.28*r: 64.820766
Z=0.29*r: 63.600337
Z=0.30*r: 62.383766
Z=0.31*r: 61.171200
Z=0.32*r: 59.962785
Z=0.33*r: 58.758671
Z=0.34*r: 57.559008
Z=0.35*r: 56.363948
Z=0.36*r: 55.173646
Z=0.37*r: 53.988257
Z=0.38*r: 52.807939
Z=0.39*r: 51.632852
Z=0.40*r: 50.463158
Z=0.41*r: 49.299020
Z=0.42*r: 48.140607
Z=0.43*r: 46.988086
Z=0.44*r: 45.841630
Z=0.45*r: 44.701412
Z=0.46*r: 43.567611
Z=0.47*r: 42.440406
Z=0.48*r: 41.319981
Z=0.49*r: 40.206523
Z=0.50*r: 39.100222
Z=0.51*r: 38.001272
Z=0.52*r: 36.909872
Z=0.53*r: 35.826224
Z=0.54*r: 34.750534
Z=0.55*r: 33.683013
Z=0.56*r: 32.623877
Z=0.57*r: 31.573348
Z=0.58*r: 30.531653
Z=0.59*r: 29.499023
Z=0.60*r: 28.475698
Z=0.61*r: 27.461923
Z=0.62*r: 26.457950
Z=0.63*r: 25.464039
Z=0.64*r: 24.480457
Z=0.65*r: 23.507481
Z=0.66*r: 22.545397
Z=0.67*r: 21.594499
Z=0.68*r: 20.655094
Z=0.69*r: 19.727498
Z=0.70*r: 18.812040
Z=0.71*r: 17.909065
Z=0.72*r: 17.018927
Z=0.73*r: 16.142001
Z=0.74*r: 15.278676
Z=0.75*r: 14.429361
Z=0.76*r: 13.594485
Z=0.77*r: 12.774500
Z=0.78*r: 11.969883
Z=0.79*r: 11.181139
Z=0.80*r: 10.408804
Z=0.81*r: 9.653448
Z=0.82*r: 8.915683
Z=0.83*r: 8.196163
Z=0.84*r: 7.495595
Z=0.85*r: 6.814744
Z=0.86*r: 6.154444
Z=0.87*r: 5.515611
Z=0.88*r: 4.899253
Z=0.89*r: 4.306497
Z=0.90*r: 3.738607
Z=0.91*r: 3.197025
Z=0.92*r: 2.683413
Z=0.93*r: 2.199721
Z=0.94*r: 1.748288
Z=0.95*r: 1.332001
Z=0.96*r: 0.954555
Z=0.97*r: 0.620943
Z=0.98*r: 0.338510
Z=0.99*r: 0.119862