A chord with a length of #5 # runs from #pi/4 # to #pi/2 # radians on a circle. What is the area of the circle?

Question

A chord with a length of #5 # runs from #pi/4   # to #pi/2   # radians on a circle. What is the area of the circle?

Caroline Aucoin · Accepted Answer

134.066

As shown in the figure ,the length of the chord AB is 5, which runs from #pi/4# to #pi/2#. This would lie in the 1st Quadrant. Now divide the circumference of the circle in eight equal parts , from 0 to#pi/4#, #pi/4# to #pi/2#, #pi/2# to #(3pi)/4#, #(3pi)/4# to #pi#, #pi# to #(5pi)/4#, #(5pi)/4# to #(3pi)/2#, #(3pi)/2# to #(7pi)/4# and #(7pi)/4# to #2pi#. The chord lengths joining these points on the circumference would all be equal to 5. If all these points are joined the resultant figure would be a regular octagon with side 5, as shown in the figure along side.

Now, let the radius of the circle be 'r'. In triangle OAB, angle O is #360/8# or #45^o#. The base angles A and B would thus be each #67 1/2 # degrees.

Draw perpendicular OD from O to side AB. This would bisect AB because OAB is an isosceles triangle. This means AD=BD= 2.5. Since ODB is a rt triangle #r cos67 1/2 = 2.5# . This would give #r= 2.5 sec 67 1/2#

= 6.53

Area of circle would be #pi r^2= 3.14 (6.53)^2#= 134.066

Maverick Smith · Answer

undefined The area of the circle can be calculated using the formula for the area of a sector of a circle. Given that the chord length is 5 and it subtends an angle of $\frac{\pi}{4}$ to $\frac{\pi}{2}$ radians, we can calculate the radius using the formula $r = \frac{c}{2\sin(\frac{\theta}{2})}$, where $c$ is the chord length and $\theta$ is the angle subtended by the chord.

Once we have the radius, $r$, we can use the formula for the area of a circle, $A = \pi r^2$, to find the area of the circle.

First, calculate the radius using the given chord length and angle subtended. Then, use the radius to find the area of the circle.