A chord with a length of #24 # runs from #pi/3 # to #(5 pi )/6 # radians on a circle. What is the area of the circle?

2 Answers
Oct 3, 2017

The area of the circle is #2304/pi# or #733.386#

Explanation:

So, a circle has an internal angle of 2#pi# radians. First we need to figure out the angle of the chord we have.

#(5pi)/6-pi/3=(5pi)/6-(2pi)/6=(3pi)/6=pi/2#

This menas that #pi/2# represents a length of 24. In order to know the circumference of the circle we need to know the length represented by #2pi#. We can get this by multiplying #pi/2# by 4, so the circumference is 96. Remember the formula for the circumference is:

#C=2pi*r#
#96=2pi*r#
#r=96/(2pi)#
#r=48/pi#

Finally, we want the area of the cicle, which is given by the following:
#A=pi*r^2#

substituting r into the equation, we get:

#A=pi(48/pi)^2#
#A=48^2/pi=2304/pi=733.386#

Oct 3, 2017

Area of circle#=905.14#

Explanation:

#theta=((5pi)/6)-(pi/3)=(5pi-2pi)/6=3pi/6=pi/2=90#degrees
#theta/2=45# deg
#sin(theta/2)=# opp. side / hypotenuse = (chord/2)/radius
#sin45=(24/2)/r#
#r=12/sin45=12/(1/sqrt2)=12sqrt2#
Area of the circle #=pir^2=(22*12sqrt2*12sqrt2)/7#
#=(22*288)/7=905.14#