Math

QuestionAn arc is 16\frac{1}{6} of a circle's circumference and measures 3π3 \pi inches. Find the circle's area in square inches.

Studdy Solution

STEP 1

Assumptions1. The length of the arc is 16\frac{1}{6} of the circumference of the circle. . The length of the arc is 3π3 \pi inches.
3. The area of a circle is given by the formula Area = \pi r^, where rr is the radius of the circle.
4. The circumference of a circle is given by the formula Circumference=πrCircumference =\pi r, where rr is the radius of the circle.

STEP 2

First, we need to find the circumference of the circle. We can do this by using the given relationship between the length of the arc and the circumference.
Circumference=Arclength16Circumference = \frac{Arc\, length}{\frac{1}{6}}

STEP 3

Now, plug in the given value for the arc length to calculate the circumference.
Circumference=3π16Circumference = \frac{3 \pi}{\frac{1}{6}}

STEP 4

Calculate the circumference of the circle.
Circumference=3π×6=18πCircumference =3 \pi \times6 =18 \pi

STEP 5

Now that we have the circumference, we can find the radius of the circle. We can do this by rearranging the formula for the circumference.
Radius=Circumference2πRadius = \frac{Circumference}{2\pi}

STEP 6

Plug in the value for the circumference to calculate the radius.
Radius=18π2πRadius = \frac{18 \pi}{2\pi}

STEP 7

Calculate the radius of the circle.
Radius=18π2π=9Radius = \frac{18 \pi}{2\pi} =9

STEP 8

Now that we have the radius, we can find the area of the circle. We can do this by using the formula for the area of a circle.
Area=πr2Area = \pi r^2

STEP 9

Plug in the value for the radius to calculate the area.
Area=π×92Area = \pi \times9^2

STEP 10

Calculate the area of the circle.
Area=π×92=81πArea = \pi \times9^2 =81 \piThe area of the circle is 81π81 \pi square inches.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord