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Calculate the perimeter of a circle.

Mathematics

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Here are the solutions to problems 3 and 4:

3. Calculate the perimeter of a circle.
To calculate a specific numerical perimeter (circumference) of a circle, the radius or diameter must be known. Without this information, we can only provide the general formula using the given value of π\pi.

  1. State the formula for the circumference of a circle.

    The circumference CC of a circle can be calculated using the radius rr or the diameter dd:

    C=2πrC = 2\pi r

    or

    C=πdC = \pi d

  2. Substitute the given value of π\pi.

    The problem specifies using π=317\pi = 3\frac{1}{7}.
    Converting the mixed number to an improper fraction:

    π=317=(3×7)+17=21+17=227\pi = 3\frac{1}{7} = \frac{(3 \times 7) + 1}{7} = \frac{21 + 1}{7} = \frac{22}{7}

  3. Provide the general formula with the specified π\pi.

    Substituting π=227\pi = \frac{22}{7} into the formulas:

    C=2×227×r=447rC = 2 \times \frac{22}{7} \times r = \frac{44}{7} r

    or

    C=227dC = \frac{22}{7} d

    Since no specific radius or diameter is provided, the perimeter cannot be calculated as a numerical value.

    4. Calculate the circumference of a circle with the following radii or diameters:
    We will use π=227\pi = \frac{22}{7} for all calculations.

    • a) r=14 cmr = 14 \text{ cm}

  4. Use the formula for circumference with radius.

    $$
C = 2\pi r
$$

  5. Substitute the values for π\pi and rr.

    $$
C = 2 \times \frac{22}{7} \times 14 \text{ cm}
$$

  6. Simplify the expression.

    $$
C = 2 \times 22 \times \frac{14}{7} \text{ cm}
$$
$$
C = 2 \times 22 \times 2 \text{ cm}
$$
$$
C = 88 \text{ cm}
$$
The circumference is $\boxed{\text{88 cm}}$.
    • b) r=4 cmr = 4 \text{ cm}

  7. Use the formula for circumference with radius.

    $$
C = 2\pi r
$$

  8. Substitute the values for π\pi and rr.

    $$
C = 2 \times \frac{22}{7} \times 4 \text{ cm}
$$

  9. Simplify the expression.

    $$
C = \frac{44}{7} \times 4 \text{ cm}
$$
$$
C = \frac{176}{7} \text{ cm}
$$
The circumference is $\boxed{\frac{176}{7} \text{ cm}}$.
    • c) d=21 cmd = 21 \text{ cm}

  10. Use the formula for circumference with diameter.

    $$
C = \pi d
$$

  11. Substitute the values for π\pi and dd.

    $$
C = \frac{22}{7} \times 21 \text{ cm}
$$

  12. Simplify the expression.

    $$
C = 22 \times \frac{21}{7} \text{ cm}
$$
$$
C = 22 \times 3 \text{ cm}
$$
$$
C = 66 \text{ cm}
$$
The circumference is $\boxed{\text{66 cm}}$.
    • d) d=7 cmd = 7 \text{ cm}

  13. Use the formula for circumference with diameter.

    $$
C = \pi d
$$

  14. Substitute the values for π\pi and dd.

    $$
C = \frac{22}{7} \times 7 \text{ cm}
$$

  15. Simplify the expression.

    $$
C = 22 \text{ cm}
$$
The circumference is $\boxed{\text{22 cm}}$.
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