Circumference Of A Circle Using 22/7 For Pi

🔑 Key Concepts

• The circumference of a circle is the one-dimensional distance around the circle.
• If the diameter is given, use \(C=\pi d\).
• If the radius is given, use \(C=2\pi r\).
• Using \(\frac{22}{7}\) for \(\pi\) produces an approximation, so use the symbol \(\approx\) after substitution.
• Rename mixed numbers as improper fractions before multiplying.
• Reduce common factors before multiplying whenever possible.
• Write circumference answers in linear units rather than square units.

✏️ Worked Examples — Using \(\frac{22}{7}\) for \(\pi\)

Example 1 — Diameter of \(1\frac{1}{6}\) ft

Find the circumference using \(\frac{22}{7}\) for \(\pi\).

The diameter is given, so use:

\[ C=\pi d \]

Substitute \(\frac{22}{7}\) for \(\pi\) and \(1\frac{1}{6}\) for \(d\).

\[ C\approx\frac{22}{7}\left(1\frac{1}{6}\right) \]

Rename the mixed number as an improper fraction.

\[ 1\frac{1}{6}=\frac{7}{6} \]

\[ C\approx\frac{22}{7}\cdot\frac{7}{6} \]

Reduce the common factor of \(7\).

\[ C\approx\frac{22}{1}\cdot\frac{1}{6} \]

\[ C\approx\frac{22}{6} \]

\[ C\approx\frac{11}{3} \]

Rename the improper fraction as a mixed number.

\[ \boxed{C\approx3\frac{2}{3}\text{ ft}} \]

Example 2 — Radius of \(2\frac{5}{8}\) ft

Find the circumference using \(\frac{22}{7}\) for \(\pi\).

The radius is given, so use:

\[ C=2\pi r \]

Substitute \(\frac{22}{7}\) for \(\pi\) and \(2\frac{5}{8}\) for \(r\).

\[ C\approx2\cdot\frac{22}{7}\left(2\frac{5}{8}\right) \]

Rename the mixed number as an improper fraction.

\[ 2\frac{5}{8}=\frac{21}{8} \]

\[ C\approx2\cdot\frac{22}{7}\cdot\frac{21}{8} \]

Reduce \(2\) with \(8\), and reduce \(21\) with \(7\).

\[ C\approx1\cdot22\cdot\frac{3}{4} \]

\[ C\approx\frac{66}{4} \]

\[ C\approx\frac{33}{2} \]

Rename the improper fraction as a mixed number.

\[ \boxed{C\approx16\frac{1}{2}\text{ ft}} \]

Example 3 — Diameter of \(8\frac{3}{4}\) ft

Find the circumference using \(\frac{22}{7}\) for \(\pi\).

The diameter is given, so use:

\[ C=\pi d \]

Substitute \(\frac{22}{7}\) for \(\pi\) and \(8\frac{3}{4}\) for \(d\).

\[ C\approx\frac{22}{7}\left(8\frac{3}{4}\right) \]

Rename the mixed number as an improper fraction.

\[ 8\frac{3}{4}=\frac{35}{4} \]

\[ C\approx\frac{22}{7}\cdot\frac{35}{4} \]

Reduce \(35\) and \(7\).

\[ C\approx\frac{22}{1}\cdot\frac{5}{4} \]

\[ C\approx\frac{110}{4} \]

\[ C\approx\frac{55}{2} \]

Rename the improper fraction as a mixed number.

\[ \boxed{C\approx27\frac{1}{2}\text{ ft}} \]

🧠 Math Vocabulary

• Circumference: The distance around a circle.
• Radius: The distance from the center of a circle to its edge.
• Diameter: The distance across a circle through its center; it is twice the radius.
• Pi: The constant ratio of a circle’s circumference to its diameter.
• Approximation: A value close to an exact value; \(\frac{22}{7}\) is an approximation of \(\pi\).
• Improper fraction: A fraction whose numerator is greater than or equal to its denominator.
• Mixed number: A whole number and a proper fraction written together.

💡 Main Idea

To find circumference using \(\frac{22}{7}\) for \(\pi\), first determine whether the radius or diameter is given. Choose \(C=\pi d\) or \(C=2\pi r\), substitute the measurements, rename mixed numbers as improper fractions, reduce common factors, and write the approximate circumference in linear units.

📚 What You Should Already Know

Students should know the difference between radius and diameter, understand that \(d=2r\), multiply and simplify fractions, and rename mixed numbers as improper fractions.

🚀 What Comes Next

Students will solve more challenging circumference problems, apply circle formulas in real-world situations, compare approximations of \(\pi\), and find a missing radius or diameter when the circumference is known.

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