A chord with a length of #35 # runs from #pi/8 # to #pi/2 # radians on a circle. What is the area of the circle?

1 Answer
Mar 5, 2016

Draw two radii to the edges of the chord to complete a triangle Then find the ratio between the angles, and apply the sine rule. Then calculate the radius and the area. (#992.2 " units"^2#)

Explanation:

  1. Draw two radii to the edges of the chord complete a triangle. You'll have an isosceles triangle with sides #r#, #r#, and 35, and angles #x#, #x#, and #theta#.

  2. The edges of the chord are #pi/8 and pi/2#, so the angle between the two radii #(theta) = pi/2 - pi/8 = {3pi}/8 "rad"#.

  3. The sum of all angles in a triangle #= 180^"o" = pi " rad"#
    #:. x + x+ theta = pi#
    #2x + {3pi}/8 = pi#
    #x = {5pi}/16#

  4. In any trangle #side_1:side_2:side_3# = #sinangle_1: sinangle_2: sinangle_3#

#:. r:r:35= sinx:sinx:sintheta#

#r:r:35= sin({5pi}/16):sin({5pi}/16):sin({3pi}/8) ~~ 0.83:0.83:0.92#

#r rarr 0.83#
#35 rarr 0.92#
#r =(35*0.83)/0.92#
#r= 31.5 " units"#

  1. The area#= pi*r^2=pi*(31.5 )^2#
    #=992.2 " units"^2#