Question

The circumference of a circle is 9 pi ft. What is the area, in square feet? Express your answer in terms of pi.

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Answer 1

`\large\boxed{\tt Area = 20.25 \pi \ ft^(2)}`

Explanation:

`\textsf{We are asked for the area of a circle.}`

`\textsf{Note that we are given only the circumference of the circle, which means that we}`

`\textsf{have to find more information from only the Circumference, which is possible.}`

`\large\underline{\textsf{What is Circumference?}}`

`\textsf{Circumference is considered the perimeter of the circle. Because a Circle has arcs}`

`\textsf{instead of sides, the Circumference is the sum of all the arcs' lengths.}`

`\underline{\textsf{How are we able to find the Circumference?}}`

`\textsf{The Circumference is measured by using the radius, or diameter of a circle, then}`

`\textsf{multiplying such by pi. Mathematicians discovered that Pi is always the measure}`

`\textsf{of the arcs when Diameter is used. Pi is an irrational number mainly used for circles.}`

`\underline{\textsf{Formulas for Circumference;}}`

`\tt C = (Diameter) * \pi`

`\tt C = 2(Radius) * \pi`

`\textsf{*The Radius is multiplied by 2 to equal the length of the Diameter.}`

`\textsf{Our goal is to find the Radius of the circle, let's use the formula including the Radius.}`

`\large\underline{\textsf{Solving for the Radius;}}`

`\tt 9 \pi = 2(Radius) \pi`

`\textsf{Let's start by using the Division Property of Equality for pi.}`

`\tt (9 \pi)/(\pi) = (2(Radius) \pi)/(\pi)`

`\tt 9 = 2(Radius)`

`\textsf{Use the Division Property of Equality again to find the Radius.}`

`\tt (9)/(2) = (2 (Radius))/(2)`

`\large\boxed{\tt Radius = 4.5 ft.}`

`\textsf{Now that we have the Radius, we are able to find the area of the circle.}`

`\large\underline{\textsf{What is Area?}}`

`\textsf{Area is the space that the surface of a shape occupies.}`

`\underline{\textsf{How are we able to find the area of a circle?}}`

`\textsf{Of course, Pi is incl\textsf{u}ded to find the area of a circle as well. As mentioned before,}`

`\textsf{we found the Radius with the Circumference in order to find the Area. The Radius}`

`\textsf{is squared, then multiplied by Pi to equal the area.}`

`\underline{\textsf{Formula for Area;}}`

`\tt Area = \pi(Radius)^(2)`

`\textsf{We know the Radius, hence we may begin solving for the Area.}`

`\large\underline{\textsf{Solving for the Area;}}`

`\tt Area = \pi(4.5)^(2)`

`\textsf{Let's follow the rule of Operations, where we should evaluate the exponents first.}`

`\tt 4.5 * 4.5 = 20.25.`

`\large\underline{\textsf{Hence;}}`

`\large\boxed{\tt Area = 20.25 \pi \ ft^(2)}`