Question

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?

Answer 1

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`