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The diagram shows a circle inside a square of side 12 cm - OCR - GCSE Maths - Question 8 - 2023 - Paper 4

Question 8

The diagram shows a circle inside a square of side 12 cm. Work out the percentage of the square that is shaded. You must show your working.

Worked Solution & Example Answer:The diagram shows a circle inside a square of side 12 cm - OCR - GCSE Maths - Question 8 - 2023 - Paper 4

Step 1

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Answer

To find the area of the square, we can use the formula for the area of a square, which is given by:

ext{Area}_{ ext{square}} = ext{side}^2

Substituting the side length:

ext{Area}_{ ext{square}} = 12^2 = 144 ext{ cm}^2

Step 2

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Answer

The radius of the circle can be calculated from the side of the square, since the circle fits perfectly inside. The radius is:

ext{radius} = rac{12}{2} = 6 ext{ cm}

Now, using the formula for the area of a circle:

ext{Area}_{ ext{circle}} = ext{pi} imes ext{radius}^2

We can substitute the radius:

ext{Area}_{ ext{circle}} = ext{pi} imes 6^2 = ext{pi} imes 36 ext{ cm}^2 \\ ext{(Approximation: } 3.14)

Thus,

ext{Area}_{ ext{circle}} ext{ (using 3.14)} ightarrow 3.14 imes 36 ext{ cm}^2 = 113.04 ext{ cm}^2

Step 3

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Answer

The shaded area can be calculated by subtracting the area of the circle from the area of the square:

ext{Area}_{ ext{shaded}} = ext{Area}_{ ext{square}} - ext{Area}_{ ext{circle}}

Substituting the values:

ext{Area}_{ ext{shaded}} = 144 - 113.04 = 30.96 ext{ cm}^2

Step 4

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Answer

To find the percentage of the square that is shaded, we use the formula:

ext{Percentage} = rac{ ext{Area}_{ ext{shaded}}}{ ext{Area}_{ ext{square}}} imes 100 ext{\%}

Now substituting the areas:

ext{Percentage} = rac{30.96}{144} imes 100 ext{\%} \rightarrow 21.5 ext{\%}

Thus, the percentage of the square that is shaded is approximately 21.5%.