Question

which of the following statements are true? select two correct answers. select two correct answers.
□ π is the ratio of the circumference of a circle to its diameter.
□ the area of a circle is the distance around the circle.
□ π is the ratio of the area of a circle to its diameter.
□ π is the ratio of the circumference of a circle to its radius.
□ the circumference of a circle is the distance around the circle.

Explanation:

1. Recall the definition of \(\pi\): \(\pi\) is defined as the ratio of the circumference (\(C\)) of a circle to its diameter (\(d\)), i.e., \(\pi=\frac{C}{d}\). So the statement " \(\pi\) is the ratio of the circumference of a circle to its diameter" is correct.
2. Recall the definition of circumference: The circumference of a circle is the distance around the circle. So the statement "The circumference of a circle is the distance around the circle" is correct.
3. Analyze the incorrect statements:

• The area of a circle is not the distance around the circle (the distance around is circumference), so "The area of a circle is the distance around the circle" is incorrect.
• \(\pi\) is not the ratio of the area of a circle to its diameter (the formula for the area of a circle is \(A = \pi r^{2}=\frac{\pi d^{2}}{4}\), so \(\frac{A}{d}=\frac{\pi d}{4}

eq\pi\) in general), so " \(\pi\) is the ratio of the area of a circle to its diameter" is incorrect.

• \(\pi\) is the ratio of circumference to diameter, not radius (since \(d = 2r\), \(\frac{C}{r}=\frac{C}{\frac{d}{2}} = 2\pi\)), so " \(\pi\) is the ratio of the circumference of a circle to its radius" is incorrect.

Answer:

• \(\boldsymbol{\pi}\) is the ratio of the circumference of a circle to its diameter.
• The circumference of a circle is the distance around the circle.