Question

1. find the circumference and diameter of these two circles. to see if they are similar, divide the circumference and diameter to prove they have a constant ratio.

Explanation:

Identify given radii

We are given two circles:

• Small blue circle: radius \(r_1 = 3\)
• Large red circle: radius \(r_2 = 5\)

Calculate diameters

Using the Diameter and Radius concept:

• Small circle diameter:

$$d_1 = 2 \cdot r_1 = 2 \cdot 3 = 6$$

• Large circle diameter:

$$d_2 = 2 \cdot r_2 = 2 \cdot 5 = 10$$

Calculate circumferences

Using the Circumference Formula Application and Circumference of a Circle concepts:

• Small circle circumference:

$$C_1 = 2\pi r_1 = 2\pi(3) = 6\pi \approx 18.85$$

• Large circle circumference:

$$C_2 = 2\pi r_2 = 2\pi(5) = 10\pi \approx 31.42$$

Calculate ratios

Divide each circle's circumference by its diameter to find the ratio:

• Small circle ratio:

$$\frac{C_1}{d_1} = \frac{6\pi}{6} = \pi \approx 3.14$$

• Large circle ratio:

$$\frac{C_2}{d_2} = \frac{10\pi}{10} = \pi \approx 3.14$$

Verify similarity

Since both circles have the constant ratio \(\frac{C}{d} = \pi\), they are geometrically similar.

Answer:

• Small Circle (radius = 3):
• Diameter: \(6\)
• Circumference: \(6\pi\) (or approximately \(18.85\))
• Ratio \(\frac{C}{d}\): \(\frac{6\pi}{6} = \pi\) (or approximately \(3.14\))

• Large Circle (radius = 5):
• Diameter: \(10\)
• Circumference: \(10\pi\) (or approximately \(31.42\))
• Ratio \(\frac{C}{d}\): \(\frac{10\pi}{10} = \pi\) (or approximately \(3.14\))

• Conclusion:
• Both circles have a constant ratio of \(\pi\), proving that all circles are similar.