A chord with a length of #1 # runs from #pi/8 # to #pi/3 # radians on a circle. What is the area of the circle?

Question

A chord with a length of #1   # runs from #pi/8   # to #pi/3   # radians on a circle. What is the area of the circle?

Sophia Alfred · Accepted Answer

Area of the circle #A_c = pi r^2 = pi (1.5555)^2 = color(brown)(7.6013)# sq. units

#/_(AOM) = theta = (pi/3) - (pi/8) = ((5pi)/24)#

#theta / 2 = ((5pi)/48)#

Length of chord # = AB = L_c = 1# Given

#OA = r = (L_c / 2)* (1 / sin (theta/2)) =( 1/2) * (1/ sin ((5pi)/48))#

#r = 1.5555 units#

Area of the circle #A_c = pi r^2 = pi (1.5555)^2 = color(brown)(7.6013)# sq. units

Noah Atkinson · Answer

undefined To find the area of the circle, we first need to determine the radius of the circle using the chord length and the central angle formed by the chord.

Given that the chord length is 1 unit and the central angle spans from π/8 to π/3 radians, we can find the radius using the chord's midpoint and the formula for calculating the radius of a circle with respect to a chord and its central angle.

The central angle θ is the difference between the two angles π/3 and π/8: 
θ = π/3 - π/8 = 5π/24 radians.

Now, using the formula for the radius (r) in terms of the chord length (l) and the central angle (θ):
r = (l / 2) / sin(θ / 2).

Substituting the given values:
r = (1 / 2) / sin(5π/48).

With the radius (r) known, we can calculate the area of the circle using the formula:
Area = π * r^2.

Substituting the calculated value of r:
Area = π * ((1 / 2) / sin(5π/48))^2.