How to Find the Area of a Shaded Part of a Square With a Circle in the Middle

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A common beginning geometry problem is calculating the area of standard shapes such as squares and circles. An intermediate step in this learning process is combining the two shapes. For instance, if you draw a square and then draw a circle inside the square so that the circle touches all four sides of the square, you can determine the total area outside the circle within the square.

    Calculate the area of the square first by multiplying its side length, ​s​, by itself:

    \text{area} = s^2

    For example, suppose the side of your square is 10 cm. Multiply 10 cm × 10 cm to get 100 cm2.

    Calculate the circle's radius, which is half the diameter:

    \text{ radius} = \frac{1}{2} \text{ diameter}

    Because the circle fits entirely inside the square, the diameter is 10 cm. The radius is half the diameter, which is 5 cm.

    Calculate the area of the circle using the equation:

    \text{area} = πr^2

    The value of pi (π) is 3.14, so the equation becomes 3.14 × 5 cm2. So you have 3.14 × 25 cm squared, equaling 78.5 square centimeters.

    Subtract the area of the circle (78.5 cm2) from the area of the square (100 cm2) to determine the area outside the circle, but still within the square. This becomes

    100 \text{ cm}^2 - \text{ cm}^2 = 21.5 \text{ cm}^2

    Warnings

    • A common mistake in this problem is to use the circle's diameter in the area equation and not the radius. Be careful to make sure you have all the correct information before you start working.

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