Question

3 select all of the true statements. a. π is the area of a circle of radius 1. b. π is the area of a circle of diameter 1. c. π is the circumference of a circle of radius 1. d. π is the circumference of a circle of diameter 1. e. π is the constant of proportionality relating the diameter of a circle to its circumference. f. π is the constant of proportionality relating the radius of a circle to its area.

Explanation:

Step1: Recall the formula for the area of a circle

The formula for the area of a circle is \(A = \pi r^{2}\).

• For option A: If \(r = 1\), then \(A=\pi(1)^{2}=\pi\).
• For option B: If \(d = 1\), then \(r=\frac{d}{2}=\frac{1}{2}\), and \(A=\pi(\frac{1}{2})^{2}=\frac{\pi}{4}

eq\pi\).

Step2: Recall the formula for the circumference of a circle

The formula for the circumference of a circle is \(C = 2\pi r=\pi d\).

• For option C: If \(r = 1\), then \(C = 2\pi(1)=2\pi

eq\pi\).

• For option D: If \(d = 1\), then \(C=\pi(1)=\pi\).

Step3: Recall the proportionality relationships

• The formula \(C=\pi d\) shows that \(\pi\) is the constant of proportionality relating the diameter \(d\) of a circle to its circumference \(C\). So option E is correct.
• The formula \(A=\pi r^{2}\) shows that the relationship between \(A\) and \(r\) is not a simple proportionality (\(A\) is proportional to \(r^{2}\), not \(r\)), so option F is incorrect.

Answer:

A. \(\pi\) is the area of a circle of radius 1.
D. \(\pi\) is the circumference of a circle of diameter 1.
E. \(\pi\) is the constant of proportionality relating the diameter of a circle to its circumference.