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K12 LibreTexts

6.6: Circle Area

  • Page ID
    2186
  • 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 \(A=\pi r^2\).

    f-d_234e9cc5f3aaa0ca982d52fb5dddb075217c3bc5093884bec6cc41fa+IMAGE_TINY+IMAGE_TINY.png
    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.

    Solution

    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?

    Solution

    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?

    f-d_c26733aea6e50bb17d9547a5b63b58264c222e53afab57cdc0494afc+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Solution

    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.

    Solution

    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 \(36\pi \).

    Solution

    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.

    Review

    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.

    Vocabulary

    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