A chord with a length of #15 # runs from #pi/3 # to #pi/2 # radians on a circle. What is the area of the circle?

Question

A chord with a length of #15     # runs from #pi/3    # to #pi/2   # radians on a circle. What is the area of the circle?

James Atkins · Accepted Answer

The area of the circle is #=2638u^2#

The angle subtended at the center of the circle is

#theta=pi/2-pi/3=1/6pi#

The length of the chord is

#l=2*r*sin(theta/2)#

So,

#r=l/(2sin(theta/2))#

#=15/(2sin(1/12pi))#

#=29#

The area of the circle is

#A=pi*r^2=pi*29^2=2638u^2#

Jayden Jackson · Answer

undefined To find the area of the circle, we first need to find its radius using the chord length and the angle it subtends at the center.

The formula to find the radius (r) of a circle given the chord length (c) and the angle subtended by the chord (θ) at the center is:

r = (c / 2) / sin(θ/2)

Given:
Chord length (c) = 15 units
Angle subtended by the chord (θ) = π/2 - π/3 = π/6 radians

Now, substitute these values into the formula:

r = (15 / 2) / sin(π/12)

Next, calculate the sine of π/12:

sin(π/12) ≈ 0.2588

Now, substitute this value into the formula and solve for r:

r ≈ (15 / 2) / 0.2588 ≈ 28.867 units

Now that we have the radius of the circle (approximately 28.867 units), we can use the formula for the area of a circle:

Area = π * r^2

Substitute the value of r into the formula:

Area ≈ π * (28.867)^2 ≈ 2615.945 square units

Therefore, the area of the circle is approximately 2615.945 square units.