TO DIVIDE A STRAIGHT ANGLE (180 degree ) INTO FIVE EQUAL PARTS.
Given:
Angle CBC’ = 180 degree
Required:
To divide the given angle CBC’ into five equal parts.
Construction:
At first, the given angle CBC’ is bisected once to get 90 degree, twice to get 45
degree, thrice to get angle ABC = 22.5 degree
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From BC, 5 equal parts (suitable and convenient) BD, DE, EF, FG, and GH are cut
off. With center B and radius BD, an arc is drawn which intersects AB at M .
Similarly, with same center B and radius BH, another arc ( which is actually half
circle HRSTUC’ ) is also drawn which intersects AB at N. From the bigger arc HS,
taking MD as an arc, 8 equal parts HI, IJ, JK, KL, LN, NP, PQ and QR are cut off. If
BR is joined and extended to A’, then angle A’BC is 1/5th of 180 degree. Taking
HR as an arc, 5 equal parts viz.HR, RS,ST,TU and UC’ are cut off. BU, BT, BS are
joined to get the required 5 equal parts.
Logic:
Let angle ABC= θ and BD = r,
Then, BH=5 * BD = 5r
Arc DM = rθ and arc HN = 5rθ
Hence, arc HN is 5 times arc DM
Again, arc HN = 5rθ, then, each part of the arc HN i.e. ,
Arcs IH = IJ = JK = KL = LN = 5 r θ / 5 = r θ
And radius of each arc = 5 r
Hence, angle subtended by each small arc at the center B = angle IBH= r θ / 5 r
= θ/5
Here, angle ABC= 180 /8 =22.5 degree, hence, angle subtended at the centre ,
i.e., angle IBC = 22.5/5= 4.5 degree. Taking 8 graduations, we will get (4.5*8)
=36 degree, which means angle RBH = angle A’BC = 1/5th of 180 degree.
Therefore, angle RBH= angle RBS = angle SBT= angle TBU= angle
UBC’=180/5=36degree
Following the same process, we can divide any angle into any no. of equal parts.
Remarks: To get best and accurate result, θ /e*n should be equal to 4.5degree ( Magic
value ). Here, θ is the given angle, ‘e’ is the no of bisected portions, i.e. 2, 4, 8, or 16 and
‘n’ is the no. of divisions. This Magic value, as calculation reveals , is that angle when
subtended at the centre by an arc , the chord length and arc length are almost equal.
Also, note that, no. of graduations taken at a time = no of bisected portions i.e., 2,4,8 or
16.
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Calculation:
When the angle subtended at the centre = 4.5 degree = pi/180*4.5 radian =pi/40 radian
The difference between arc length and chord length = pi/40 *R– 2Rsin2.25degree
=R( 0.07854 -2* 0.03926)
=R( 0.07854- 0.07852)
=0.00002 R = almost nil
SHYAMAL KUMAR DAS
Address:
Flat no: 1C /15
~~~~~~~``` Uttarpara Housing Estate
88B, G.T.Road. Bhadrakali
Uttarpara. Pin:712232
West Bengal. India.
Ph: 033-2664-5991
Mob: 09432861354
Date: 09-09-2011
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