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6.6: Circle Area

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    Area of a Circle

    To find the area of a circle, all you need to know is its radius. If r is the radius of a circle, then its area is \(A=\pi r^2\).

    Figure \(\PageIndex{1}\)

    We will leave our answers in terms of \(\pi\), unless otherwise specified.

    What if you were given the radius or diameter of a circle? How could you find the amount of space the circle takes up?

    Example \(\PageIndex{1}\)

    Find the area of a circle with a diameter of 12 cm.


    If \(d=12\text{ cm }\), then \(r=6\text{ cm }\). The area is \(A=\pi (6^2)=36\pi \text{ cm }^2\).

    Example \(\PageIndex{2}\)

    If the area of a circle is \(20\pi \text{ units }\), what is the radius?


    Plug in the area and solve for the radius.

    \(\begin{aligned} 20\pi &=\pi r^2 \\ 20&=r^2 \\ r&=\sqrt{20}=2\sqrt{5}\text{ units } \end{aligned}\)

    Example \(\PageIndex{3}\)

    A circle is inscribed in a square. Each side of the square is 10 cm long. What is the area of the circle?

    Figure \(\PageIndex{2}\)


    The diameter of the circle is the same as the length of a side of the square. Therefore, the radius is 5 cm.

    \(A=\pi 5^2=25\pi \text{ cm }^2\)

    Example \(\PageIndex{4}\)

    Find the area of the shaded region from Example 3.


    The area of the shaded region would be the area of the square minus the area of the circle.

    \(A=102−25\pi =100−25\pi \approx 21.46\text{ cm }^2\)

    Example \(\PageIndex{5}\)

    Find the diameter of a circle with area \(36\pi \).


    First, use the formula for the area of a circle to solve for the radius of the circle.

    \(\begin{aligned}A&=\pi r^2 \\ 36\pi &=\pi r^2 \\ 36&=r^2 \\ r&=6\end{aligned}\)

    If the radius is 6 units, then the diameter is 12 units.


    Fill in the following table. Leave all answers in terms of \(\pi\).

    radius Area circumference
    1. 2
    2. \(16\pi\)
    3. \(10\pi\)
    4. \(24\pi\)
    5. 9
    6. \(90\pi\)
    7. \(35\pi\)
    8. \(7\pi\)
    9. 60
    10. 36

    Find the area of the shaded region. Round your answer to the nearest hundredth.

    1. f-d_6d85c3ece461a1ac5632843ecefc2dcf4d967d68ca471890e9082248+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{3}\)
    2. f-d_3b6f51be8181d0779c9b529800eabc3d11f4de07c97852c4e82e592a+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{4}\)
    3. f-d_c41bd33249e1cff70119e9dcabb75781e5eacd8ada186fba0f3b023f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{5}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 10.10.


    Term Definition
    chord A line segment whose endpoints are on a circle.
    circle The set of all points that are the same distance away from a specific point, called the center.
    circumference The distance around a circle.
    diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
    pi (or \(\pi\) ) The ratio of the circumference of a circle to its diameter.
    radius The distance from the center to the outer rim of a circle.

    Additional Resources

    Interactive Element

    Video: Determine the Area of a Circle

    Activities: Area of a Circle Discussion Questions

    Study Aids: Circumference and Arc Length Study Guide

    Practice: Circle Area

    Real World: Ringside Seats

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