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SectionA.5Circle

FigureA.5.1Circle

A circle's diameter is always twice the length of its radius. A circle's perimeter is called circumference. Deriving a circle's circumference and area formulas will be discussed in later math classes. For now, we will simply learn how to use them:

Circle's Circumference

\(C=\pi d=2\pi r\text{,}\) where \(C\) stands for a circle's circumference, \(d\) for diameter, and \(r\) for radius.

Circle's Area

\(A=\pi r^2\text{,}\) where \(A\) stands for a circle's area, and \(r\) for radius.

The constant \(\pi\)'s value is \(3.1415926\ldots\text{.}\) A scientific calculator has a \(\pi\) button, so we don't have to type its value digit by digit.

ExampleA.5.2

A circle's diameter is \(10\) meters. Find the following values.

  1. Find the circle's circumference. Leave the result in terms of \(\pi\text{.}\)

  2. Find the circle's circumference. Round the result to two decimal places.

  3. Find the circle's area. Leave the result in terms of \(\pi\text{.}\)

  4. Find the circle's area. Round the result to two decimal places.

Solution

  1. We will use a circle's circumference formula:

    \begin{align*} C\amp=\pi d\\ \amp=\pi(10\text{ m})\\ \amp=10\pi\text{ m} \end{align*}

    The circle's circumference is \(10\pi\) meters.

  2. We will use a circle's circumference formula:

    \begin{align*} C\amp=\pi d\\ \amp=(3.1415926\ldots)(10\text{ m})\\ \amp\approx31.42\text{ m} \end{align*}

    The circle's circumference is approximately \(31.42\) meters.

  3. To find a circle's area, we need to find its radius, which is half of its diameter. Since its diameter is \(10\) meters, its radius is \(5\) meters. We will use a circle's area formula:

    \begin{align*} A\amp=\pi r^2\\ \amp=\pi(5\text{ m})^2\\ \amp=25\pi\text{ m}^2 \end{align*}

    The circle's area is \(25\pi\) square meters.

  4. We will use a circle's area formula:

    \begin{align*} A\amp=\pi r^2\\ \amp=(3.1415926\ldots)(5\text{ m})^2\\ \amp\approx78.54\text{ m}^2 \end{align*}

    The circle's area is \(78.54\) square meters.

CheckpointA.5.3