In a circle centered at point $O$, the ratio of the area of sector $AOB$ to the area of the circle is $\frac{3}{5}$. What is the approximate measure, in radians, of the central angle corresponding to $\widehat{AB}$? Round the answer to two decimal places.
Answer (2)
Define the ratio of the sector's area to the circle's area as 5 3 .
Express the area of the sector as 2 1 r 2 θ and the area of the circle as π r 2 , where θ is the central angle in radians.
Set up the equation 2 π θ = 5 3 and solve for θ , obtaining θ = 5 6 π .
Approximate the value of θ to two decimal places: 3.77 .
Explanation
Problem Analysis We are given that the ratio of the area of sector A OB to the area of the circle is 5 3 . We need to find the measure of the central angle corresponding to arc A B in radians, rounded to two decimal places.
Define variables Let A sec t or be the area of sector A OB , and A c i rc l e be the area of the circle. Let θ be the central angle in radians corresponding to arc A B . We are given that A c i rc l e A sec t or = 5 3 .
Area formulas The area of a sector is given by A sec t or = 2 1 r 2 θ , where r is the radius of the circle. The area of a circle is given by A c i rc l e = π r 2 . Therefore, A c i rc l e A sec t or = π r 2 2 1 r 2 θ = 2 π θ .
Solve for theta We have 2 π θ = 5 3 . Solving for θ , we get θ = 5 3 × 2 π = 5 6 π .
Approximate theta Now, we approximate the value of θ . θ = 5 6 π ≈ 3.7699111843077517 . Rounding to two decimal places, we get θ ≈ 3.77 radians.
Examples
Imagine you're cutting a pizza, and you want one slice to be exactly 3/5 of the whole pizza. The angle of that slice, measured from the center, would be the central angle we just calculated. This concept is useful in many real-life situations, from dividing resources proportionally to designing circular structures.
The central angle corresponding to arc AB in the circle is approximately 3.77 radians, derived from the ratio of the area of sector AOB to the area of the circle being 5 3 . This was calculated using the formulas for the area of a sector and a circle. After simplification, the angle was found to be 5 6 π radians, which approximates to 3.77 radians.
;
