Question

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?

Answer 1

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`

Answer 2

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`