A circle has a circumference of 31pi centimeters. What is the diameter, in centimeters, of the circle?

1. TRANSLATE the problem information

• Given information:
  • Circumference = 31π centimeters
  • Need to find: diameter in centimeters

• This means we can write: \(\mathrm{C = 31\pi}\)

2. INFER the approach

• We need to use the circumference formula to find radius first
• Since \(\mathrm{C = 2\pi r}\), we can substitute our known circumference
• Then we'll use the relationship that diameter = 2r

3. TRANSLATE into mathematical equation

Set up the equation using the circumference formula:

\(\mathrm{31\pi = 2\pi r}\)

4. SIMPLIFY to solve for radius

Divide both sides by π:

\(\mathrm{31\pi \div \pi = 2\pi r \div \pi}\)

\(\mathrm{31 = 2r}\)

5. INFER the final answer

• We found that \(\mathrm{2r = 31}\)
• Since diameter = 2r, the diameter is 31 centimeters

Answer: 31

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse the circumference formula and use \(\mathrm{C = \pi r}\) instead of \(\mathrm{C = 2\pi r}\), or they might use \(\mathrm{C = \pi d}\) but then solve incorrectly.

If they use \(\mathrm{C = \pi r}\), they would set up: \(\mathrm{31\pi = \pi r}\), leading to \(\mathrm{r = 31}\), then \(\mathrm{diameter = 2(31) = 62}\). This leads to confusion since 62 isn't typically among answer choices.

Second Most Common Error:

Poor INFER reasoning: Students correctly find that \(\mathrm{2r = 31}\) and \(\mathrm{r = 15.5}\), but then give 15.5 as their final answer instead of recognizing they need \(\mathrm{diameter = 2r = 31}\).

This causes them to provide the radius instead of the diameter, missing the key requirement of the question.

The Bottom Line:

This problem tests whether students can distinguish between radius and diameter while working with the circumference formula. The elegant part is that the algebra works out so that 2r equals the final answer directly, but students must recognize this connection.