Select the correct answer.
Points $A$ and $B$ lie on a circle centered at point $O$. If $\frac{\text { length of } \widehat{A B}}{\text { radius }}=\frac{\pi}{10}$ what is the ratio of the area of sector $A O B$ to the area of the circle?
A. $\frac{1}{10}$
B. $\frac{\pi}{10}$
C. $\frac{1}{20}$
D. $\frac{\pi}{20}$
Answer (2)
Calculate the angle θ subtended by the arc A B at the center O using the formula θ = radius arc length . In this case, θ = 10 π .
Determine the area of sector A OB using the formula 2 1 r 2 θ , which gives 20 π r 2 .
Calculate the area of the entire circle using the formula π r 2 .
Find the ratio of the area of sector A OB to the area of the circle: 20 1 .
Explanation
Find the angle subtended by the arc AB at the center O. Let r be the radius of the circle. The length of the arc A B is given by 10 π r . The angle θ (in radians) subtended by the arc A B at the center O is given by θ = radius arc length = r 10 π r = 10 π .
Calculate the area of sector AOB and the area of the circle. The area of sector A OB is given by 2 1 r 2 θ = 2 1 r 2 ( 10 π ) = 20 π r 2 . The area of the circle is given by π r 2 .
Find the ratio of the area of sector AOB to the area of the circle. The ratio of the area of sector A OB to the area of the circle is π r 2 20 π r 2 = 20 π π = 20 1 . Therefore, the ratio of the area of sector A OB to the area of the circle is 20 1 .
State the final answer. The ratio of the area of sector A OB to the area of the circle is 20 1 .
Examples
Imagine you're cutting a pizza into slices. The ratio of the area of one slice (sector) to the whole pizza (circle) can be calculated using the angle of the slice and the radius of the pizza. This is useful in determining how much of the pizza each slice represents. For example, if the arc length of your pizza slice is 10 π times the radius, then the slice's area is 20 1 of the whole pizza's area. This concept applies to various scenarios where you need to determine proportions of circular areas, such as in engineering, design, or even cooking!
The ratio of the area of sector AOB to the area of the circle is 20 1 . Therefore, the correct answer is option C. This is determined by calculating the angle subtended by the arc and using it to find both the area of the sector and the area of the circle for comparison.
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