A chord with a length of #4 # 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 #4    # runs from #pi/3    # to #pi/2   # radians on a circle. What is the area of the circle?

Savannah Aguilar · Accepted Answer

The area is #24^2/pi# From #pi/3# to #pi/2# there are #pi/6# radians. Then #r pi/6 =l# where #r# is the circle radius and #l# the chord. From this relationship we obtain the radius #r = 6 l/pi# Equating now the circle area #A = pi r^2 = pi(6*4/pi)^2 = 24^2/pi#

Sadie Allen · Answer

undefined To find the area of the circle, we need to first determine its radius. The chord length of 4 units is the diameter of the circle, since a chord passing through the center of the circle is a diameter. Thus, the radius is half of this length, which is 2 units.

The angle subtended by the chord is $ \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} $ radians. This angle is the central angle of the circle sector defined by the chord.

The area of the circle sector can be calculated using the formula:

$$ \text{Area of sector} = \frac{1}{2} \times \text{radius}^2 \times \text{central angle} $$

Substituting the given values, we get:

$$ \text{Area of sector} = \frac{1}{2} \times 2^2 \times \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} $$

Therefore, the area of the circle is $ \frac{2\pi}{3} $ square units.