A chord with a length of #12 # runs from #pi/12 # to #pi/6 # radians on a circle. What is the area of the circle?
2 Answers
Area of a circle is
Explanation:
Picture above reflects the conditions set in the problem. All angles (enlarged for better understanding) are in radians counting from the horizontal X-axis
We have to find a radius of a circle in order to determine its area.
We know that chord
Construct an altitude
Consider a right triangle
We know that cathetus
Therefore, hypotenuse
Knowing radius, we can find an area:
Let's express this without trigonometric functions.
Since
we can express the area as follows:
Another trigonometric identity:
Therefore,
Now we can represent the area of a circle as
Another approach same result
Explanation:
The chord AB of length 12 in the above figure runs from
So polar coordinate of A
Applying distance formula for polar coordinate
the length of the chord AB,
So area of the circle