Question

6. A sector of a circle is a region bound by an arc and the two radii that share the arc's endpoints. Suppose you have a dartboard that has a diameter of 20 in and it is divided into 20 congruent sectors. Find the area of one sector.
Part I: Find the central angle. (Hint: A circle has 360 degrees.) (1 point)


Part II: Use your answer from Part I to find the fraction of the circle that one sector will take up. (1 point)


Part III: Use the fractional part from Part II with the area formula to find the area of one sector of the circle to the nearest tenth. (2 points)

Answer 1

Given the dartboard of diameter $20in$, divided into 20 congruent sectors,

• The central angle is $18^\circ$

• The fraction of a circle taken up by one sector is $\frac{1}{20}$

• The area of one sector is $15.7in^2$ to the nearest tenth

The area of a circle is given by the formula

$A=\pi r^2$

A sector of a circle is a fraction of a circle. The fraction is given by $\frac{\theta}{360^\circ}$. Where $\theta$ is the angle subtended by the sector at the center of the circle.

The formula for computing the area of a sector, given the angle at the center is

$A_s=\dfrac{\theta}{360^\circ}\times \pi r^2$

Given information

We given a circle (the dartboard) with diameter of $20in$, divided into 20 equal(or, congruent) sectors

Part I: Finding the central angle

To find the central angle, divide $360^\circ$ by the number of sectors. Let $\alpha$ denote the central angle, then

$\alpha=\dfrac{360^\circ}{20}\\\\\alpha=18^\circ$

Part II: Find the fraction of the circle that one sector takes

The fraction of the circle that one sector takes up is found by dividing the angle a sector takes up by $360^\circ$. The angle has already been computed in Part I (the central angle, $\alpha$). The fraction is

$f=\dfrac{\alpha}{360^\circ}\\\\f=\dfrac{18^\circ}{360^\circ}=\dfrac{1}{20}$

Part III: Find the area of one sector to the nearest tenth

The area of one sector can be gotten by multiplying the fraction gotten from Part II, with the area formula. That is

$A_s=f\times \pi r^2\\=\dfrac{1}{20}\times3.14\times\left(\dfrac{20}{2}\right)^2\\\\=\dfrac{1}{20}\times3.14\times10^2=15.7in^2$

Learn more about sectors of a circle brainly.com/question/3432053