25 Area and circumference of Circles
Critical Thinking
Solve each problem.

Suppose you know that the radius of one circle is three times the radius of another circle.
Explain how you can determine the ratio of the areas of the circles and the ratio of the
circumferences of the circles without knowing the actual measurements.

Question

25 Area and circumference of Circles
Critical Thinking
Solve each problem.

Suppose you know that the radius of one circle is three times the radius of another circle.
Explain how you can determine the ratio of the areas of the circles and the ratio of the
circumferences of the circles without knowing the actual measurements.

UpStudy AI · Accepted Answer

To determine the **ratio of the areas** and the **ratio of the circumferences** of two circles when the radius of one circle is three times the radius of the other, you can use the fundamental formulas for the area and circumference of a circle. Here's a step-by-step explanation:

### 1. **Understanding the Given Information:**
- Let’s denote the radius of the smaller circle as $$ r $$.
- Therefore, the radius of the larger circle is $$ 3r $$ (since it's three times the smaller radius).

### 2. **Ratio of the Areas:**
- **Formula for Area of a Circle:** 
  $$
  A = \pi r^2
  $$
- **Area of the Smaller Circle ($$ A_1 $$):**
  $$
  A_1 = \pi r^2
  $$
- **Area of the Larger Circle ($$ A_2 $$):**
  $$
  A_2 = \pi (3r)^2 = \pi \times 9r^2 = 9\pi r^2
  $$
- **Ratio of the Areas ($$ \frac{A_2}{A_1} $$):**
  $$
  \frac{A_2}{A_1} = \frac{9\pi r^2}{\pi r^2} = 9
  $$
- **Conclusion:** The ratio of the areas of the two circles is **9:1**.

### 3. **Ratio of the Circumferences:**
- **Formula for Circumference of a Circle:**
  $$
  C = 2\pi r
  $$
- **Circumference of the Smaller Circle ($$ C_1 $$):**
  $$
  C_1 = 2\pi r
  $$
- **Circumference of the Larger Circle ($$ C_2 $$):**
  $$
  C_2 = 2\pi (3r) = 6\pi r
  $$
- **Ratio of the Circumferences ($$ \frac{C_2}{C_1} $$):**
  $$
  \frac{C_2}{C_1} = \frac{6\pi r}{2\pi r} = 3
  $$
- **Conclusion:** The ratio of the circumferences of the two circles is **3:1**.

### **Summary:**
- **Area Ratio:** 9:1
- **Circumference Ratio:** 3:1

These ratios are derived directly from the properties of circles and the given relationship between their radii. Importantly, you don't need the actual measurements of the radii to determine these ratios; knowing the proportion between the radii is sufficient.
The ratio of the areas of the two circles is 9:1, and the ratio of their circumferences is 3:1.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Geometry Questions