Question

select all quantities that equal π.
a the circumference of a circle of radius 1
b the circumference of a circle of diameter 1
c the diameter of a circle of radius 1
d the diameter of a circle of circumference 1
e the area of a circle of radius 1
f the area of a circle of diameter 1
g the constant of proportionality relating the diameter of a circle to its radius
h the constant of proportionality relating the diameter of a circle to its circumference

Explanation:

Step1: Recall the formulas

• Circumference formula \(C = 2\pi r=\pi d\) (where \(r\) is radius and \(d\) is diameter).
• Area formula \(A=\pi r^{2}=\frac{\pi d^{2}}{4}\).
• Relationship between diameter and radius \(d = 2r\), so \(d=kr\) where \(k = 2\).
• Relationship between circumference and diameter \(C=\pi d\), so \(C=md\) where \(m=\pi\).

Step2: Analyze each option

• Option A:

If \(r = 1\), then \(C=2\pi\times1=2\pi
eq\pi\).

• Option B:

If \(d = 1\), then \(C=\pi\times1=\pi\).

• Option C:

If \(r = 1\), then \(d = 2\times1=2
eq\pi\).

• Option D:

If \(C = 1\), then \(d=\frac{C}{\pi}=\frac{1}{\pi}
eq\pi\).

• Option E:

If \(r = 1\), then \(A=\pi\times1^{2}=\pi\).

• Option F:

If \(d = 1\), then \(r=\frac{1}{2}\), and \(A=\pi\times(\frac{1}{2})^{2}=\frac{\pi}{4}
eq\pi\).

• Option G:

Since \(d = 2r\), the constant of proportionality \(k = 2
eq\pi\).

• Option H:

Since \(C=\pi d\), the constant of proportionality \(m=\pi\).

Answer:

B. the circumference of a circle of diameter 1, E. the area of a circle of radius 1, H. the constant of proportionality relating the diameter of a circle to its circumference