Question

How do you find the radian measure of the central angle of a circle of radius 6 inches that intercepts an arc of length 27 inches?

Answer 1

Hint: In this problem, we have to find the radian measure of the central angle of the circle of radius 6 inches that intercept the arc of length 27 inches. We know that the given radius of the circle is 6, we also know that the circumference of the circle, by using the circumference of the circle formula, we can find radian measure that intercepts the arc of length 27 inches.

Complete step by step answer:
We know that the circumference of the circle is.
Circumference of the circle = \[2\times \pi \times r\]…... (1)
We also know that the given radius, r = 6.
Now we can substitute the value of the radius, r in the formula (1), we get
\[\Rightarrow 2\times \pi \times r=12\pi \]
We got \[12\pi \] inches, we know that it subtends the central angle of \[2\pi \] radian.
But we have to find the radian measure that intercepts an arc of length 27 inches, so 27 can be divided by the circumference of the circle, which subtends a central angle of \[2\pi \] radian, we get
\[\Rightarrow \dfrac{27}{12\pi }\times 2\pi \]
Now we can cancel the similar terms and convert the fraction part to decimal to get the radian value.
\[\Rightarrow \dfrac{27}{6}=4.5\] radians.
Therefore, the radian measure of the central angle of a circle of radius 6 inches that intercepts an arc of length 27 inches is 4.5 radians.

Note:
Students make mistakes in finding the value of radian measure, students should understand what is being asked for, to use the formula. Students make mistakes in dividing the given total length of 27 inches by the circumference of the circle.